Tennis scoring looks arbitrary until you see the shape of it: points build games, games build sets, sets build a match, and at every level you have to win by a clear margin.
Points
Within a game, the score goes love (zero), 15, 30, 40, game.
Win four points and you win the game, but only if you are two points clear. At 40-40 the score is deuce. From deuce, the next point gives a player advantage. Win the following point and you take the game; lose it and the score returns to deuce. This can repeat indefinitely.
The server's score is always called first. "Thirty-fifteen" means the server has thirty.
Games and sets
Win six games and you win the set, again with a two-game margin. So 6-4 is a set and 6-5 is not. At 6-5 the match continues; if the trailing player wins the next game to make 6-6, a tiebreak is played.
The tiebreak
In a tiebreak, players count in ordinary numbers: one, two, three. First to seven points wins, with a two-point margin, so 7-5 wins it and 7-6 does not. You keep playing until somebody leads by two.
Service alternates in a fixed pattern: one player serves the first point, then service changes every two points. Players change ends every six points. The tiebreak winner takes the set 7-6.
Match length
Most matches are best-of-three sets. Men's singles at the four Grand Slam tournaments is best-of-five.
Final sets used to be played out without a tiebreak, which produced famous marathons. All four Grand Slams now use a ten-point tiebreak at six games all in the deciding set: first to ten points, win by two.
Why the numbers are so odd
The most-repeated explanation is a clock face, with quarter past, half past, and forty-five shortened in speech to forty. Nobody can prove it, and the honest answer is that the system was inherited from real tennis and survived because players were used to it. It is tradition, not logic.
Terms you will hear
A break of serve is winning a game in which your opponent served, the fundamental currency of a tennis match, because holding your own serve is expected. Break point is the point that would achieve it. Set point and match point are the equivalents at the higher levels.
A bagel is a set won 6-0. A love game is a game won without conceding a point.
The short version
Four points win a game, by two. Six games win a set, by two, with a tiebreak at 6-6. Two or three sets win the match. Every level requires a margin, which is why tennis has no clock and no guaranteed ending.