What does the second number (in this case the 10) refer to when I play live chess?

10 seconds are added to your time after each move.

You get 15 minutes for the game, plus 10 additional seconds on your clock every time you make a move.

For x|y: you get x min. and y additional seconds.

10 extra seconds added to your clock per move you make. Assuming an average game of 40 moves, thats 400 seconds = 6 minutes and change above your base 15 minutes.

If you havent figured that, consider quitting chess altogether.

LoL