I have a 97 Honda Accord and when I started it in the morning and put it in reverse gear, it didn't move (no, the hand brakes were not on). I gave a little bit more gas, but then I heard some clicking noise in front right tire area.

If I were to diagnose it myself, how do I go about?

I took the tire out, but now I have no clue how to diagnose it.

Any ideas?

 

Try this
1) with handbrake on put car in neutral jack up BOTH front tires in the air
2) try to spin the right front tire the the driver side, should spin pretty easily if not the caliper piston could be stuck all the way out.
3) why does your title say when brakes are pressed?

 

Axle popped out?