501 and 301

 Most professional matches are "501 up".  This is the simplest of dart games.  Each player starts with a score of 501 and takes turns to throw 3 darts.  The score for each turn is calculated and deducted from the players total.  Bullseye scores 50, the outer ring scores 25 and a dart in the double or treble ring counts double or treble the segment score.  The objective is to be the first player to reduce the score to exactly zero, the only caveat being that the last dart thrown must land in a double or the bullseye.
If a player reduces the score to 1 or goes below zero, the score is bust, that turn ends immediately and the score is returned to what it was at the start of that turn.  e.g. if a player has 32 to go out and the first dart is a 16, the second is a 15, the player is bust and the score is returned to 32.  So on the last turn, it is not necessary to throw all 3 darts - a player can win with the first or second dart of the turn.
Because a player who misses a targeted double is likely to score the single of that segment, good players attempt to leave themselves with a repeatably bisectable number such as 24 or most ideally 32 - double 16. So for instance, if a player has double 16 left, and hits a 16, he has double 8 left and if he then hits an 8 he has double 4 left and so on - this is advantageous because no extra darts need to be thrown in order to reduce the score to an even number... It so often happens that people reduce their score to 1 (typically while aiming for double 1), some people play a very unofficial rule called "splitting the 11". This rule says that when the score is reduced to 1, instead of going bust, the player must "split the 11" by throwing a dart between the two numbers forming the number 11 on the edge of the board. This is tricky.
301 is played in exactly the same way but players start the game with 301.  Some games require a double to start scoring as well as to finish the game.
When the score goes below 171, professional players know the numbers to aim at order to check out in a single turn with 3 darts. Here is a table of such combinations. Some scores have multiple check out options but in most cases, we only show the best one:
170
  
t20, t20, bull. (Highest checkout possible.)
167
  
t20, t19, bull.
164
  
t20, t18, bull.
(preferable to t19, t19, bull because it doesn't require two darts in one bed and a miss keeps the number even.)
161
  
t20. t17, bull.
160
  
t20, t20, d20.
158
  
t20, t16, bull or
t20, t20, d19
157
  
t20, t19, d20
156
  
t20, t20, d18
155
  
t20, t15, bull or
t20, t19, d19
154
  
t20, t18, d20
153
  
t20, t19, d18
152
  
t20, t20, d16
151
  
t20, t17, d20
150
  
t20, t18, d18
(preferable to bull, bull, bull unless you are in the mood for intimidating your opponent!)
149
  
t20, t19, d16
148
  
t20, t16, d20
( followed by t18, t18, d20)
147
  
t20, t17, d18 or
t19, t18, d18
146
  
t20, t18, d16
(followed by t19, t19, d16).

For all numbers lower than 146, there are at least 3 check-out combinations.