What is the "increment" in chess? I saw this in many online games, but I do not understand how it works. With every move we can gain time, but what is it dependent on how much we get? Can please somebody explain the system behind it.
Take a look at Wikipedia's time control article:

Fischer delay (or time increment), invented by Bobby Fischer. When it becomes a player's turn to move, the delay (or increment) is added to the player's remaining time. For example, if the delay is five seconds and the player has ten minutes remaining on his clock when his clock is activated, he now has ten minutes and five seconds remaining.

And in the usage of chess servers:

On chess servers (like Internet Chess Club and Free Internet Chess Server) the time control is usually expressed with a formula: time+increment. Where time is a fixed amount of time (in minutes) assigned at move 0 for the entire game, and increment (in seconds) is the amount added after each move.
When the increment is 0 the time control turns into a sudden death: so a 5+0 game is a game with sudden death at 5 minutes.
When the increment is bigger than 0 the time control is a Fischer delay: e.g. a 2+12 is a game with Fischer delay with 2 minutes + 12 seconds each move.
A game can even have a time set to 0: the clock is set to 0 and started after the first move with the increment only on both sides, and each move each player is granted only the increment: a 0+10 is a game where after the first move each player has 10 seconds to move.


To sum it up:
Time increment gets added to your current time to compensate for a probable delay (for humans is the reaction time and for servers may compensate for latency etc.). If you are fast enough to play your move before the increment ends, the extra time gets added to your current time thus augmenting your remaining time.