While all styles of golf scoring use the same method of counting how many strokes a player has shot, the way a winner is determined based on those totals changes depending on the system. Match play is the most common variation from standard stroke play scoring.
Stroke play is the most basic form of golf scoring. On every hole, players record the number of strokes, with penalties, it took to complete the hole. At the end of the round, each player adds up the scores for each hole, generally 18. The player who needed the fewest shots to complete the round wins.
Match play rewards players for shooting lower than their opponent, but the scale is smaller, focusing on a hole-by-hole basis. If both players shoot the same score on a hole, nobody wins, and neither player receives a point. If a player shoots lower than his opponent, whether by 1 stroke or 5, he wins the hole. At the end of the round, the player who wins the most holes wins the match. A match can also end early when a player is ahead by more holes than are left to play.
A major difference between stroke and match play is the size of field you're competing against. In stroke play, players usually compete against every player on the course, so simply outscoring the players in your group doesn't ensure victory. In match play, you play only one opponent at a time, meaning large tournaments with multiple rounds and the winners of each match moving on to a new opponent for each round.
Because match play is head-to-head competition, players have more power than they have in stroke play, where every stroke counts and affects everybody in the tournament, not just a single opponent. In match play, you may concede at any point. A concession can be a small act, such as saying an opponent is close enough that he gets credit for making his final putt without taking it. Or a player can simply concede a hole, saying there is no way he can win it and giving his opponent the point.
Match play rewards aggressive play slightly more than a stroke competition. For example, a player attempting to shoot at a flag over water, rather than a safe shot over grass to the fat of the green, risks two strokes (penalty stroke and replaying the shot) by going for the flag, with a payoff of possibly one-putting instead of two-putting. In stroke play, this is risking two shots to gain one, meaning the player must be fairly sure he can make the shot. In match play, a player is risking a chance to win the hole against a chance to lose it, an even tradeoff, making it a higher percentage play.