Master recordings are the record of a musical performance that can be played back, or reproduced.
You can make money off of master recordings by selling your own recordings directly to your audience (for example, make your own CDs and sell them at your shows, or use CD Baby to distribute your master recordings digitally through music stores like iTunes), or by signing a deal with a record label that will help you make your master recordings and sell them to the public.
The composition side - also referred to as the publishing side - is the underlying musical composition of a piece of music; the music and lyrics.
Music publishing refers to the money you make from government mandated royalties that must be paid to you when other people or companies use music that you wrote (the composition). That can refer to your record company making a copy of your music (mechanical royalties), when a radio station, website, or restaurant plays your music to help them make money (public performance royalties), or if a television show or film uses your music in their productions (sync royalties).