Most players know about Stroke Play - every shot is counted and added up and the player with the lowest score of all wins. But Match Play (the oldest form of golf) is not as commonly understood. Perhaps the following explanation will help.
In a USGA amateur Championship, there are two days of stroke play qualifying to determine exactly the 64 players who will make it to match play. The "Match Play Tree" is then established -- much like a tennis tournament or NCAA basketball - and players are seeded according to how they played during stroke play.
Match play is a competition played by holes rather than total strokes for the round. In USGA amateur Championships, two opponents play against each other and while there may be other players on the course, each group is its own match and has nothing to do with the rest of the field. The winners of each match keep advancing until there is only one player left. With 64 players, this occurs after 6 matches.
For example, let's look at the imaginary match between players A and B below. A match always starts at "All Square," that is, the match is even, no one has an advantage or disadvantage. A wins the 1st hole, so is "1 up." After A wins the 2nd hole, A is then "2 up." (It doesn't matter how many strokes the hole is won by, no more than "1 up" can be the result of the scores from any one hole.)
The players halve the 3rd hole, so there is no change in the status of the match. B then wins the 4th hole, which leaves A only 1 up. B wins the 5th hole, so the match returns to All Square ("AS"). B then wins the 6th hole, and takes the lead 1 up. And so on.
Notice that a score does not have to be recorded in match play (see the "x" on the 6th hole for A). The result of the hole (won, loss, or halved) simply needs to be determined. In fact, "conceding" is allowed. Player A, for example, can concede the 6th hole to B without finishing it. Players may also concede that their opponents will hole out with their next strokes; therefore, if B wants to concede A's one foot putt on the 7th hole for a 4, B can - and A doesn't have to putt.
The match goes on in this fashion until one player is leading by a greater number than the number of holes left to be played. For example, if B is 5 up with 4 holes left to play, the match is over as A can not possibly come back. B is said to have won the match, "5 and 4." If the players are still All Square after the 18th hole, the match is continued hole by hole until a winner is determined. So, if A and B play the 1st and 2nd holes again, halving both, and A wins the 3rd hole, A is said to have won the match, "21 Holes."
We hope this will assist in your understanding of match play and specifically the method of scoring that is used. Please contact the USGA Rules Department with any additional match play questions.