Write some music. Perhaps you are adept at guitar or piano. Compose your own tunes. Write some lyrics if you want. If you aren't instrumentally inclined, however, several types of software let you create your own tunes (such as Audacity or Garage Band). You can also use these programs to create remixes of known songs.
Record yourself. If you want to sink the money into a recording session, the professional quality is worth it. But you can also take video of yourself for sharing or record your music onto your computer.
Publish the music. Post a video of yourself making music on YouTube. A quick Google search will yield a plethora of sites that host music. MySpace is one perennial favorite. You can upload an MP3 for others to hear or download.
Promote. Throw your music all over social media. Tweet links to your music and post it on Facebook. Go to blogs and forums and let people know about your music. The squeaky wheel gets the recording contract.