In blitz, an increment is a small unit of time added after each move. So in a 3-minute game, you might choose to play '3+2' which means two seconds are added to your allocation each time you move. Over a 30-move game, you'd therefore gain an extra minute.
This model can be used OTB as well. In some major tournaments, instead of the standard 40 moves in 120 mins, you might often find 40 in 80 minutes + 1 min increment. In other words, you gain a minute every time you move; over 40 moves, you gain 40 mins which - lo, and behold - gives you a total of 120 mins.
So if it comes to the same thing, why bother with an increment-based method? The answer is that, with the increment, violent time scrambles are avoided at the end of the game because you always have a minute in which to move.
Does this make life easier? Yes, but not necessarily for the players! It makes life far easier in major tournaments for the arbiters - they no longer have to oversee dozens of time scrambles.
Of course, increments as described above are only available OTB with digital clocks; or at blitz, via online facilities