Why does the sun make me feel warm? For a while I thought that the reason I felt warmth from the sun was because my skin was being hit by photons, but then I realized that photons also hit me when I take an X-ray, but I don't feel any heat from that. So, why is it that you feel warmth from the sun, or any hot object for that matter?
 A: There's two main things to consider - energy and absorption charasteristics of different photon wavelengths.
The Sun emits a lot of energy, obviously. Even at Earth's distance from the Sun, the energy concentration is still far from negligible - when this energy impacts your body and is absorbed, it mostly causes heating (a bit complicated by wavelength, but we'll get to that). How much energy is that? Well, at rest, the adult human body radiates around 300-600 W, or 300-600 J/s. This is actually in roughly the same magnitude as the insolation energy - the average varies a lot by lattitude, mostly, and it's different at different times of the year, of course. For example, southern Spain has an average of about 200 W per square metre (more in the summer, less in the winter), while the equator has around 1000. The amount of energy you absorb depends on how much surface area you're exposing, and at what angle. But we're only dealing with ballparks here, we already have a rough magnitude.
How does the X-ray compare? Well, the X-ray machines are actually operating at an extremely low power output. According to wikipedia, 300 J is the lethal dose of X-rays - so even a single second of X-ray exposure energetically equivallent to normal sunlight would kill you! This makes it obvious that your routine X-ray scan is way, way below the kind of energy you receive from the Sun, and far too low to be detected by human senses - don't forget that everything in the room is also radiating, and at room temperature, this is again the same order of magnitude as the solar light, and your own heat radiation (this is why you don't really "feel" losing those 300 W of heat - roughly the same amount is absorbed from the environment). The X-rays are utterly insignificant in comparison.
Now, wavelength. The most important thing here is that different wavelengths of light have different behaviour when interacting with different kinds of matter. Wavelength depends on the energy of the individual photons, which in turn affects how those photons interact with matter.
Visible light is more or less the range where the individual photons have enough energy to induce chemical changes, but not enough to break the stronger bonds - this is what makes it perfect for seeing, it easily influences the light-sensitive molecules in our eyes, but doesn't destroy them (and us) utterly. It's easy to understand the principles like "opacity" with visible light - if something absorbs light, it is opaque; if it reflects light, it is "shiny"; and finally, if it doesn't interact with light, it is transparent. For example, clear glass is usually transparent and shiny, so it reflects part of the incident visible light, and absorbs very little, letting most through. Clouds tend to reflect visible light a lot, which makes them distinctly white from the top, and blocking some of the sunlight from coming to the surface (especially when the cloud gets really deep - basically, the darker a cloud appears from the bottom, the more light it reflected back up, and the deeper it is).
At lower energies lies the infrared (literally "below red", red being the lowest energy visible light). This is what we commonly call "heat radiation", because it far dominates at the low temperatures we commonly encounter - almost all of your own radiation is infrared, for example. It actually dominates even at much higher temperatures, but since those also cause the emission of visible light, we tend to take that as the more important, even though most of the energy is still emitted as infrared. The visible radiation begins around 480 °C, a dull red colour; the surface of the sun is somewhere around 6000 °C. The sunlight is energetically mostly composed of infrared (~50%) and visible light (~40%). Glass is usually opaque to infrared light - this is one of the things that makes greenhouses work; the material in the greenhouse absorbs some of the incident visible light, but when that energy is radiated back outside, it will be radiated as infrared, unable to pass through the glass. Clouds absorb near-infrared quite effectively, which is why even very shallow clouds in the summer can make you feel "cold" - the sunlight that reaches you can lose a big portion of that 50% slice mentioned earlier. Deep clouds can block almost all of the energy coming from the sun, both infrared and visible.
At higher energies we have ultraviolet ("above violet"). UV light (only a tiny fraction being "X-rays" - technically, they are ultraviolet, but are usually grouped separately), is around 8% of sunlight by energy at the top of the Earth's atmosphere, and around 4% when it finally comes through the atmosphere and the ozone layer. So in terms of the total energy flux, they are mostly negligible. Their main danger comes from their per-photon energy - they have enough energy to strip atoms of their electrons, and break even strong molecular bonds; the most important for us, they are powerful enough to damage our DNA. The lower energy ultraviolet light tends to be absorbed by clear glass quite readily, so you're not going to get tanned when sunbathing behind a window (tanning salons use different kind of glass, transparent to the "desired" band of UV light). As you get to the really high energies, you find that less and less photons are absorbed by matter - and whenever they are, they cause big changes.
X-rays are energetic enough to pass almost unhindered through the human body - basically, at those energy levels, the most important thing is density of the matter. This is what makes X-rays so useful in medicine - many X-rays pass or scatter right through us and we can use that information to build a fairly accurate image of our insides, clearly showing e.g. bones and inner organs. The drawback is that when the X-ray hits a chain of DNA, it causes damage - if there's enough X-rays absorbed (or even scattered) by the DNA, it can easily overwhelm our ability to fix the errors, causing radiation poisoning (cells not able to fix their DNA self-destruct, basically) and cancer (some of the previously mentioned cells don't self-destruct). The lethal dose is pathetically small compared to the energies we encounter every day, so much so that if you feel the heat from an X-ray source, you're pretty much guaranteed to die.
A: X-rays do warm you up. It's just that the X-rays are more dangerous per photon (they can do major damage to cells and DNA, and are known to cause tumors and cancer), so they limit the amount of time you're exposed to the bare minimum needed for a clear image.
The total energy from standing in the sunlight for several seconds is much higher than the energy from all the X-rays you're likely to take in your life, which is why you feel it but don't feel the X-ray.
Addendum:
Total Energy from an X-ray.
I found this page with radiation doses from common radiology treatments. The maximum radiation dose listed is 10 millisieverts (mSv). 1 Sv is defined as one joule (J) of energy per kilogram (kg) body mass. Assuming an average person is around 70 kg, 10 mSv corresponds to 0.7 J of energy absorbed. Note that the actual energy is probably a bit lower, because the sievert definitions also account for the biological effects (so a dose to an internal organ is weighted higher per actual joule than a dose to a finger or something). But this should be a decent approximation.
Total Energy from Sunlight.
The sun outputs about 1300 watts per square meter (W/m²) in space near the earth, which gets reduced to around 650 W/m² in the middle of the day after going through the atmosphere. 1 watt is defined as 1 joule per second (J/s). So that's about 650 joules of energy per second per square meter.
According to this site, the surface area of a human body is between 1 and 2 m². At least half of that is on the side of your body not facing the sun, and you'll get less radiation if the sun is shining on the end of your body (like your head/shoulders) instead of the front or back of your body. If we assume the sun is right overhead, and you're laying on your back facing up at the sun, then you've got 0.5 to 1 m² in the sunlight (it's a little less because some of that surface area is parallel to the sunlight, but we're in the ballpark).
Ok, combining all of the above, sunlight shines down with 650 J/(s*m²) * [0.5 to 1] m² = 325 to 650 J/s. One second of direct sunlight is therefore 465 to 930 times more energy than one X-ray image, hence why it feels so much hotter.
A: In addition to the answer from @MichaelS, you need to consider where the energy from each source is deposited:
Sunlight energy is deposited on/in the  skin where there are numerous nerve endings.  An increase in skin temperature is "measured" and your brain is aware of it.
X-ray energy which is absorbed by the body is mainly absorbed by bones and some cartilage, with very few (if any) nerve endings, so the brain is not aware of the absorption.
A: There are multiple "kinds" of photons - different wavelengths have different effects on you.
X-ray works somewhere around the 1nm range of the spectrum. It is ionizing radiation which can mostly go through soft materials but can harm cells when passing. So you usually get only the minimal needed amount of photons to create the image and not much more.
The Sun outputs many different wavelengths part of which is the infrared range. You may know that one as thermal radiation (which it is only part of, another part is the visible light).
Infrared is the range from a few µm to 1mm. It can go through some opaque materials, but not very deep in your body.
In the end you get only small amount of X-ray photons and they effectively "do something" mostly inside you - not many receptors of temperature in there. Whereas sunlight is partially absorbed by top layers of your skin where the temperature receptors "live" (the other part is reflected so others can see you + Sun emits a bit of X-ray, radio waves and other parts of spectrum too, but those are mostly filtered out by Earth atmosphere). And you get a lot of photons from sunlight because they are not so harming (not in the same numbers as X-ray but you can get sunburns too after some time - thats UV part of the spectrum mostly - just between X-ray and visible/infrared)
A: It seems like despite the lengthy responses no one has actually given you the real answer lol. A very large fraction of the sun's light output is infrared. Water tends to absorb infrared wavelengths well (as opposed to reflecting them or letting them pass through). The body is about 60% water. Therefore: the body tends to absorb a large fraction of the light emitted by the sun. That's it. 
Here you can see a graph of how well water absorbs various wavelengths. Notice the dip near the visible wavelengths - this is why you can see through water, the light passes through. Also notice how the graph goes up at the infrared wavelengths; water absorbs IR.
You can test this yourself as follows. Incandescent light bulbs (the ones with a filament) operate similarly to the sun - they emit light because the filament heats up, and when objects heat up they emit a lot of light. If you put your hand in front of an incandescent light bulb you will feel a lot of heat, because most of the light it emits is infrared. This is why incandescent bulbs are being phased out, because a lot of energy is wasted producing infrared wavelengths, which are invisible. Put your hand in front of a fluorescent light bulb and you won't feel much if any heat. They operate by a different mechanism and produce mostly just visible light, so they are much more efficient and won't heat you up much.
The total energy from an x-ray is very small, so even if you absorbed all the energy from an x-ray (which you don't, that's the point of an x-ray, to have some of the x-rays pass through you), you wouldn't feel anything.
