Typically playing songs on an MP3 player (or any music playing device such as a CD player or turntable) consists of playing individual songs with a varying amount of silence between songs/tracks. However, if you listen to Abbey Road, for example, some of the tracks are seamlessly run together. CD players and turntables manage this just fine and you don't notice any break in the music. However, if you try this with most MP3 players (and MP3 playing software) there will be a silent, but very noticable gap. The missing feature here is called "gapless playback".
Leo Davidson has written an interesting, in-depth article on gapless playback (with illustrative sound samples) and describes how common players (generally don't) deal with the problem.