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Infrared region is a part of electromagnetic spectrum that is mostly responsible for the radiative heat transfer in our everyday life. It is expressed by the fact that the peak of the Planck distribution at room temperature lies in the infrared range:

You're completely right about that all forms of electromagnetic radiation carries energy, and you can refer to Bob's answer for the technical details. It's also quite false that only infrared radiation will heat things, but there's some truth hidden in the common misconceptions, so let's break things down.

Beyond 1.5 PHz (shorter than 200 nm), atmospheric absorption increases suddenly, as the photon energy becomes high enough to ionize oxygen. At even higher frequencies, they will also interact with nitrogen.

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As black-body emission goes up by the fourth power of temperature, this starts to involve significant amounts of energy, and this is where you start feeling the presence of warm things.

As you go through this frequency range, they start to interact with human bodies more and more. You might have noticed how the signals of TV and radio are affected by your mere presence near the antenna. The amount of energy involved tends to be rather low though, and when they interact with you, the energy deposited is distributed all over your body, so you won't feel it on your skin.

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Not all that much changes as the frequency increases to 430 THz (700 nm), except the black-body temperatures go up to around 4 kilokelvin and the radiative energies involved continue increasing by the fourth power of the temperature. This is the stuff you feel when near a fire, or an incandescent lightbulb.

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I've often heard that Infrared rays are called "heat rays". However, I feel like this term is a misnomer. Don't all the wavelengths of electromagnetic radiation carry energy?

Very little of the energy of Gamma rays is absorbed by tissue, i.e., tissue is basically transparent to Gamma rays. They can even pass through several inches of lead. But as they pass though human tissue they energy that is absorbed can cause ionizations that damage tissue and DNA. For this reason, it is called ionizing radiation.

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Ultraviolet light is still mostly absorbed by the skin, but if your skin feels warm due to ultraviolet radiation, you'll get a terrible sunburn in a hurry.

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Of course, other EM radiation can heat things up too, but we rarely feel heat from them, and if we do, it's a bad thing. High-energy X-rays can heat tissue, as can microwaves, but we try to avoid such exposures. So, in everyday experience, if you feel heat coming off a hot thing (but the ambient air isn't so hot) then your skin is "seeing" bright infrared.

Some exceptions to these arguments clarify the things further: In metals at very small temperatures electronic contribution to the internal energy is larger than phononic contribution. Why? Again because of two points:

Yes, it is a bit of a misnomer, but it is informal. It's short for "the rays that we don't see, but we often feel as heat." I agree that it can cause confusion. Much of the heat we get from the sun is in the form of visible light, and so if you think of infrared as being "in charge" of transmitting heat, you'll have an inaccurate view.

Electromagnetic waves with frequencies a bit higher than infrared are visible light. Therefore, although they also carry heat, they are mainly identified with the information that we get through our eyes.

Clarification In thermodynamics/statistical mechanics heat is the energy transferred from one system to another on the microscopic level (unlike work, which is due to macroscopic changes). In this case an object is in contact with radiation. Heat rays is not a physics term, it should not be literally interpreted as "rays carrying heat". But the reason why we use this term to describe infrared radiation is the one that I give above.

Judging by how gamma rays are highly penetrating and are dangerous when absorbed by tissues, radiations of lower wavelengths should carry more energy, and should be able to increase the internal energy of the object that absorbed it much more than infrared rays can. This seems consistent with the conservation of energy for an isolated system: $$T_{ER} = \Delta E_{int}$$ where $T_{ER}$ stands for transfer of energy by electromagnetic radiation

However, you also cannot heat metals efficiently with visible light because of reflections. This has to do with typical electronic concentrations $n$ which is such that the dominant absorption of metals is in UV frequency range (plasmons) or even even lower due to $d$-$d$ transitions. With increasing the photon energy, very soon the ionization starts to dominate. Photoionized electrons can in principle heat the system, however, we are here in a completely different regime.

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Phonons are frozen (e.g. as predicted by the Debye model). The specific heat of electrons varies as $T$ (linearly with temperature, whereas the specific heat of phonons as $T^3$ (on the basis of Debye model for $T\ll T_D$, below the Debye temperature $T_D$).

"When UV rays reach your skin, they damage cells in the epidermis. In response, your immune system increases blood flow to the affected areas. The increased blood flow is what gives sunburn its characteristic redness and makes the skin feel warm to the touch."

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X-rays start 'soft', meaning they don't penetrate much, and are strongly absorbed by air, but as frequencies increase, wavelengths get shorter and photon energy increases. Around 10 keV (120 pm, 12.5 EHz), penetration depth starts to exceed 1 mm, crossing into "Hard X-Ray" territory. Hard X-Rays will penetrate deeper, allowing them to do distribute their energy beyond your skin. Again, if you feel heating effects from this, you should be worrying about the lethal dose of ionizing radiation you just received instead.

Comment on the energy conservation The radiative energy absorbed depends not only on the energy of photons of given frequency, $h\nu$, but also on the number of photons, $n_\nu$, i.e., the absorbed energy at a given frequency is $$E_\nu=n_{\nu}h\nu.$$ If we assume thermal distribution for $n_\nu$ and do calculation, we obtain the above-mentioned Planck formula. This is why the peak of the Planck formula corresponds to the ferquencies where the most heat is transferred.

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There are nice answers by @rogervadim and @bobd, and I feel like I need to add a nice example why as you say classifying only IR as heat rays is a misnomer.

All light carries energy, but in the typical human experience, only infrared and visible light will be felt as heating you significantly. The visible light will also typically be accompanied by infrared light, so it's not strange to assume that the heat is carried (only) by the infrared light.

Going up to 1 THz, the wavelength shrinks down to .3 mm (300 $\mu m$). Used mainly for high-bandwidth wireless communication and radar, this starts to enter the range of frequencies that interacts mainly with your skin, and you will actually feel. The story goes that using microwaves for heating food was discovered by a radar engineer whose chocolate bar melted when he walked in front of an antenna. That's at a very high power level though, and you won't normally encounter those outside of a microwave oven.

Yes they do, but the amount of energy that is actually absorbed depends on the frequency. Per the Hyperphysics website (http://hyperphysics.phy-astr.gsu.edu/hbase/mod3.html) regarding the interaction of radiation with matter:

In the case of X-rays and Gamma rays, it's because they don't interact with the skin in the same way as infrared, namely, they do not create a feeling of warmth on the skin.

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Going up to 750 THz, the wavelength continues to decrease to about 400 nm. Not all that much changes compared to near infrared, but there are some notable points.

But the ultimate answer is that radiative heat transfer can (and sometimes does) include other then IR wavelength light, including visible, so classifying radiative heat transfer exclusively as IR is not correct.

The difference between X-Rays and Gamma Rays are their origin: X-Rays are generated using electronic processes, while Gamma Rays are generated using nuclear processes. Their energy ranges overlap, but a typical Gamma Ray photon might be at 300 EHz (1 pm, 1.25 MeV). They behave similarly to hard X-Rays, but at higher energy, more so.

Going up to 1 GHz, we're starting to enter the realm of microwave radiation, though calling them radio waves is still correct. The wavelengths can get as short as 0.3 m (30 cm), and we're not far from the frequency of the typical microwave oven (2.45 GHz, with a wavelength of about 12.5 cm).

...radiations of lower wavelengths should carry more energy, and should be able to increase the internal energy of the object that absorbed it much more than Infrared rays can.

If you do a search on for instance "led lamp emission spectrum", you'll find more details on just what sort of radiation LED bulbs and other light sources emit.

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Here's an experiment you could easily try. Find an old-fashioned 60 watt incandescent bulb, which emits a lot of infrared along with the visible light. Place your hand a few inches/cm from it. You feel heat, don't you? Now take an LED bulb that emits the same lumens of visible light, and hold your hand at the same distance. I bet you don't feel any heat at all, right? That's because the bulb emits very little infrared.

Jumping up several orders of magnitude, up to 1 MHz, the wavelengths are still at least 300 m. Your body still doesn't really interact with it, being more than 2 orders of magnitude smaller.

Phononic contribution to the specific heat, and therefore to the internal energy of matter at normal conditions is dominant.

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At frequencies up to 1 kHz, your wavelength is at least 300 km (300 Mm/s / 1000 /s). That means your body is completely insignificant compared to the wave passing over it. It barely interacts with it at all. Interacting with these efficiently requires something with a size on the order of a planet. The main natural source of them is lightning strikes.

As the photon energy increases, the number of molecules the photons can break increases, increasing the potential for damage and sunburn, though no ultraviolet light appears to be entirely safe.

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"as you move upward in frequency from infrared to visible light, you absorb (the energy) more and more strongly. In the lower ultraviolet range, all the uv from the sun is absorbed in a thin outer layer of your skin. As you move further up into the x-ray region of the spectrum, you become transparent again, because most of the mechanisms for absorption are gone. You then absorb only a small fraction of the radiation, but that absorption involves the more violent ionization events"

Radio waves are really quite a broad term, ranging from waves of just a few Hz, up into the GigaHerz. Let's start at the low end.

Part of this range is used for thermal cameras that are intended to track high-temperature heat sources, typically the heat engines that power tanks, jets and rockets.

I think "heat rays" is a very loose and informal term, so don't take it too strictly. The usage probably stems from the fact that most objects are at a temperature that primarily emits IR. To start emitting visible it has to be very hot, like steel in a forge. And because of this, night vision or "heat vision" cameras are made to detect IR.

Also, if you use a thermal camera, you're actually measuring a specific band on infrared light, and there are several such bands depending on what you're looking for. (animals and their environment, or the exhaust of heat engines)

It is simply a matter of perception, and language. We sense (some of) the infrared spectrum as heat, while we sense the visible spectrum as light.

For our purposes, we'll define "far infrared" as everything up to 100 THz (wavelengths down to 3 $\mu m$). Like short microwaves, these will interact with your skin, and unlike microwaves, the black-body temperatures associated with these go up to about 80 degrees Celsius, beyond human body temperature.

It's because they are emitted by objects that we in daily life consider hot. Our skin is sensitive to these so we can avoid getting burned. Also we detect absense of heat waves as cold, so we can avoid hypothermia.

Ultraviolet is named such because it's beyond violet: we can't see it. As the frequencies increase, more things start to change.

I'm going to be talking about what how various frequencies (and their corresponding wavelength) interact with your body, and where they might come from.

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My favorite example is the glowing hot pieces of metal, emitting a larger spectrum of radiative heat transfer, including IR and visible. The hot metal is trying to reach thermal equilibrium with its environment in every way it can, and this will include emitting visible (other than IR of course) light too. I believe it is as you say a misnomer because in this example you can beautifully see how the object is dissipating heat including visible light, thus classifying only IR as heat rays is not correct. What is correct to say is that because our universe is fundamentally quantum mechanical, the processes involved in heat transfer are based ultimately on QM too (but certainly some can be explained classically), and one of the fundamental reasons QM was "invented", was the UV catastrophe, as you can see from the other answers, the peak of the Planck distribution lies in the IR range, and different wavelength light can interact (be absorbed) involving different QM processes.

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Visible light is also absorbed by matter, causing valence-conduction band transitions by electrons. Despite the respective photon energy is larger ($h\nu_\mathrm{viz}\gg h\nu_\mathrm{IR}$), the net effect is very small because of the difference in specific heats of electrons and phonons. This can be back-traced again to two points:

Heat is what you feel on your skin, not any energy. probably you had x- rays at some time. the have lots of energy, but did you feel heat?

Planck radiation has a maximum intensity at a wavelength that depends on the temperature of the body. For example, at room temperature (~300 K), a body emits thermal radiation that is mostly infrared and invisible. At higher temperatures the amount of infrared radiation increases and can be felt as heat, and more visible radiation is emitted so the body glows visibly red. At higher temperatures, the body is bright yellow or blue-white and emits significant amounts of short wavelength radiation, including ultraviolet and even x-rays. The surface of the sun (~6000 K) emits large amounts of both infrared and ultraviolet radiation; its emission is peaked in the visible spectrum.

However, if you walk out in the sun, and feel its 1 kW/m^2 irradiance, about half the energy heating you is actually visible light, not infrared.