APM is short for "Actions per Minute" and is an average or instantaneous measure of the rate at which a player can perform actions (including spam, see EPM)
In replays you can see how many APM a player is having in average and in real time. The APM in StarCraft II is calculated from Normal Game Speed. If you want to know the APM of a player that has played on the Faster game speed, you must therefore multiply the APM shown in the replay with 1.38. E.g. a player is shown to have 100 APM in the replay but if he played on faster game speed, it is therefore 100*1.38 = 138.
APM is sometimes used by some people as a measure of skill. This is generally incorrect, as what is known as "spamming" can highly inflate the APM for a player. Day very elegantly summarizes the disparity between APM and skill as such - high APM doesn't make anyone a better player, but better players have higher APM because they have much better game sense and awareness. He further elaborates that newer players get sucked into the "APM = skill" fallacy, and think that being good in StarCraft (or any RTS) comes down to mouse speed. Then, if they are unable to achieve a high enough mouse speed they are turned off and stop playing.
High APM (outside of spam) is the direct result of game awareness; a strong player is performing actions at several different levels such as micro-managing their units, continuing harvester production, erecting new buildings, and scouting throughout the game. Simply because the player is aware that these actions must be performed and has a sense of what needs doing, they give the impression of preternatural speed and reflex. Their APM is just a result of them doing what needs to be done. Many new players try to unnecessarily increase their APM using custom maps. It is generally suggested that the only time you need to actively increase your APM is when your brain is working faster than your hands and you can't keep up. The human ability to carry out thoughts with actions is quite impressive and you'll be surprised how sufficiently high your APM can get just by playing the game normally.