There are several common occurrences during a round of golf where a player is going to lift their ball and in some of these cases they will need to put the ball back to where it was lifted. Perhaps the most common occurrence is a ball lifted from the putting green under Rule 16-1b (Lifting and Cleaning Ball). The process of returning a ball to its previous location is called “replacing” and is covered under Rule 20-3a (Placing and Replacing).
Who can do the replacing? That depends on the form of play and, to some extent, the owner of the ball.
Before we have the need to know who can replace a player’s ball we need to understand just who can lift it. Rule 20-1 permits the player to lift his ball, but also allows the player to authorize anyone else to do so as well (but the player is on the hook that the authorized party does so properly)!
So, if a player marks the position of his ball on the putting green and lifts it, only he can replace. If he has employed a caddie and he gives the caddie permission to lift his ball that is at rest on the putting green, either the player or the caddie can replace it. If the form of play included a partner, than in both cases mentioned above, the player’s partner would also be able to replace the ball.
We have this rules animation which further explains marking, lifting and replacing a golf ball.