What is burn-in for headphones? Do you just loop music continuously or is there a special waveform to be used? Is it supposed to make them sound better?

Burn-in is the process of breaking in the headphones. Some believe it improves the SQ. Some don't. Music will work fine. Pink noise can work. But don't use test tones. And in fact, you can simply let it break-in as you listen to them. Or you can hasten the process by letting them play all the time for a few days when you are not using them.

If you do plan on burning in your headphones, I recommend you listen to a few of your favorite tracks before, and then listen to them again after the burn in process. 

I've had good results with pink noise (http://simplynoise.com).

From what I have experienced, burn-in makes a difference but depends on the type of headphones and condition. Burn-in usually will make a difference on new headphones or headphones that have been sitting idle for a long while. Some headphones may only take a few hours to notice a change in the sound signature and some can take many, many hours to fully break in. Most average around 50 hours for a noticable difference.
 
For example, I had a pair of Koss KTXPro1 that sounded absolutely awful when I first got them. I mean they sounded terrible, the SQ was so muddy and bloated to the point where I really thought they might have been fakes but I thought to myself that they are only $16 headphones and I bought them from Sonic Electronix, which is an authorized Koss dealer. So I tried to break them in but they were taking so long... However, to this day, these headphones were the ones that took the longest to fully break-in from my collection, but they are also the ones that made the biggest improvement. The KTXPro1's took over 150 hours to break in, but the difference was incredible! They went from being awful to one of the best sounding budget headphones I  have ever heard. Well for around $16 anyways, but seriously, they sound alot like the Porta Pros, which I also own and A/B'd against but the resolution on the KTX is noticeably lower than the PPs. Still very good budget phones.  
 
 
Another example is the Panasonic HTF600s. They sounded very good right out of the box, but the EQ was an extreme V shape. It had plenty of bass but the highs were peaky in the vocal range and the mids were really recessed. I logged my break-in progress and the biggest improvement for my set came between 40-50 hours, which was kind of funny because afterwards I came across some head-fi posts and it turns out many people have also experienced this 50 hour phenomenon. I thought that was a pretty neat correlation...
 
The method I used to break-in my headphones is I created a large playlist with different genres of music, then after about every 4-5 songs, which is about 20 minutes, I add about 10min of Pink Noise. I added more songs after that and sandwiched more pink noise in between, but you could probably get away with just the 20+10min of pink noise then set it on repeat, although it doesn't hurt to add more musical variety. I had set the volume to slightly louder than I could tolerate and played it continuously on a spare PC.
 
Here is a link to download pink noise and some other tracks:
 
https://archive.org/details/TenMinutesOfWhiteNoisePinkNoiseAndBrownianNoise
 
Hope this helps.

Great info offbeat, thanks.

Np.
 
Also, on some playlists I made I put a 20 min silent track after about 2.5 hours to give them a rest/settle time. You may want to consider that as well.