1. Writing on the golf ball for ID purposes
When you "mark your ball" in this sense, you write on the golf ball - letters, a smiley face, dots, whatever - for identification purposes.
Rule 6-5 states: "The responsibility for playing the proper ball rests with the player. Each player should put an identification mark on his ball."
As noted, that identification mark can be anything the player wishes. The reason for marking the ball is to ensure that there is no mix-up during play that results in golfers playing the wrong ball. Say you and your opponent are both playing Titleist Pro V1 balls with the number "3." And those balls wind up right next to each other in the fairway. Which is which?
If you and your opponent each marked his ball before teeing off, you'll be able to tell the difference.
2. Placing a ball marker on the ground before lifting the golf ball
The second usage of "mark the ball" or "marking the ball" refers to the process of denoting the golf ball's position before picking up the ball.
In most areas of the golf course (outside of the putting greens), the ball can be lifted only in special circumstances covered in the rules. On the putting green, you can pick up the golf ball for any reason. But you always must mark the ball's position before lifting it, to ensure that you replace it in the correct spot.
Golfers carry ball markers - usually a small coin or something similar - for the purpose of marking the ball on the green.
