Compare the relationship between the sounds volumewise, if they are more controlled it could mean they are being compressed.
For example in a drum group, if the snare is spiking way above the rest of the kit, it could be that the drum group is not being compressed. And when everything sounds more even, it is being compressed.
At least that's what I got from Dr. Compressor, compression goes beyond that though, but I think that's a good way to start recognizing the sound of compression.

I second what Roberto says; I also started by focusing on the snare. Whichever of the two options has a more lively snare sound (more transient and perceived loudness), that one is less compressed. A more squashed/contained snare would be the more compressed version.

I also listen for the relationship between the snare and the kick, sometimes it's also quite a clear difference.

stick to the basics, listen to the dynamics remember a compressor is usually used either to rise, lower or balance the dynamics of a sound, listen to what stands out the most and figure out how hard is X,Y,Z being attacked, or how clear is your lower end or how is it affecting your mid range, etc... Just Keep it simple! 😊

From the echoes of the toms I visualize the size of the room and the distance. So far, so good.

From the echoes of the toms I visualize the size of the room and the distance to the drummer. Compression adds room size/distance. Works for me so far.

I've found since playing Compressionist and also listening to the compressed sound vs. the Bypass sound (no compression) is instructive. Then also comparing the various answer possibilities can, over time, give you a feel for what both the different compression ratios and different attack times do to the original sound.
Note the higher the compression ratio, the more compression takes place.
Similarly, the lower the attack time, the quicker the compression kicks in, and thus the more compression takes place.

Ditto to what @John Miller said. Spend some time playing with Compressionist and mess around with the "Yours" answer settings to hear how Ratio, Attack, and Release affect the sounds. When I first tried SoundGym I was instantly hooked with Compressionist and it is still my favorite game of the lot. (Suggestion for Compressonist: set the Makeup level slider halfway between the 200ms and 300ms ticks of the Release column for a consistent level match between the "Compressed" example and the "Yours" answer.)
Also compression can be used for a myriad of different possibilities. As mentioned by @Mark Turnauckas it can bring out the room size/distance, but sometimes it can do the opposite, depending on what is needed. In Dr Compressor it is pretty common for the first half of examples to lean the compression effect in a certain way (e.g. more pronounced room ambience, or conversely less room ambience with more transients getting through) and then the second half of the examples they start leaning things in a different direction. Sneaky.
And not to overstate the obvious, the biggest thing to listen for regardless of how the room/ambience if affected is simply a reduced dynamic range of the overall sound. Not necessarily louder or softer but rather more "even" or "glued together". Sometimes this can result in things seeming louder or perhaps slightly attenuated depending on how a compressor's make-up gain is applied. With Dr Compressor it can go either way so don't trust the "louder vs softer" notion. 😊

we need jason romney to explain the compression 😂

Listen for the transients. Normally the more compressed sound will have less difference in volume between the transient and the rest of the sound. The most obvious transients to me (in order) are the snare, kick then cymbals. You can also listen for consistency in volume.

and often the Reverb is more obvious

The snare can be the biggest giveaway for me. The more natural the snare hit sounds, the less compressed (at least, on Dr. Comp's settings). If that doesn't work, try the kick - if its attack sounds like a whack, it's less compressed than an attack that sounds like a short whoop.
That's not true for all compressor settings, but for Dr. Comp it's about right. I'm finding that Compressionist is a good trainer for Dr. Comp, it teaches you what to listen for.

Listen for the attack. Mainly if you are listening to drums. Listen for a prominent kick or snare in the first version, and and compare that to your version.

Also a nice cheat, just to confirm that you hear it correctly: look at meters, maybe the meters of your audio interface, or any meter to see which sample hits higher in volume. But don't get accustomed to doing this -

To add to the great advice already given, I'd say to listen for what is the quietest part of the overall sound you're listening to. I like to try to zero in on the room sound/ambience and any tails of any sounds as they will be louder when there is more compression.

Thanks!

@Lance Clark I was struggling with Dr.compressor for a bit too. I found spending time in Compressionist helped me over the hump, simply because you can bypass the compressor and play with the settings to hear all the characteristics of the compressor.

Daniel i do the exact opposite:
i individuate the loudest element and i find out if has brought down in the other track. I could check easily the compressed track ;)

I made a video for my students yesterday that might help.

Jason, your video was of tremendous help! Now I know what to look for :)

I'm no expert, in fact, I'm struggling with the same game. I come from a background in electronic music, so listening carefully to drum overheads is definitely a change of pace.
Where I'd normally listen for compression (punch, pump, transient emphasis, and distortion) I rarely find it in this game. Instead, like other people said, listen for a change in room tone (more reverb and ring from the snare) and balance between the pieces of the kit, (in many cases, the snare vs kick). Often the more "punchy" feeling example is the *lesser* compressed of the two.

Just listen for the one that sounds more natural and real. That'll be the least compressed of the two.