Rules 15-1 and 15-2 explain that the player must complete play of the hole with the ball with which he began the hole unless he is proceeding under a Rule that permits him to substitute a ball. The player may change balls between the play of two holes as well.
Rules 26-1 (Water Hazard Rule), 27-1 (Ball Lost or Out of Bounds) and 28 (Ball Unplayable) are examples of Rules that permit the player to substitute another ball. Rule 5-3 permits a player to substitute another ball during the play of a hole when his original ball has become unfit for play during the play of that hole. Other Rules (e.g. Rule 18, Rule 24, and Rule 25-1) permit the player to substitute another ball only if the original ball is not immediately recoverable.
Note that Rule 16-1b, the Rule that allows the player to lift his ball from the putting green, does not permit the player to substitute another ball; this precludes the player from substituting a “putting ball.”
A player who substitutes another ball when not permitted to do so incurs the general penalty (loss of hole in match play or a two stroke penalty in stroke play) under the applicable Rule. For instance, a player taking relief, without penalty, under Rule 25-1 due to interference from ground under repair is required to do so with the original ball (unless it is not immediately recoverable). If that player substitutes a ball (and makes a stroke with it) when his original ball is immediately recoverable he incurs the general penalty under Rule 25-1 (as it is the applicable Rule in this case).
When changing balls, the player is permitted to substitute a ball of another brand or type unless the Committee has adopted the One Ball Condition of Competition (see Appendix I; Part C; Section 1c). This optional condition (usually referred to as ‘The One Ball Rule’) is generally adopted only in events that are limited to professional golfers or highly-skilled amateur golfers. Generally, this condition of competition is not adopted in club-level competitions.