Get inspired. This is hard to do consciously, but it's how every great song gets started. Start with a catchy or meaningful lyric that you can't get out of your head, a haunting emotion, or a melodic hook that you can't stop whistling. Get something pure and inspired to begin working with.
Play with your idea. If you play an instrument, try jamming around the melody or finding a chord progression that fits the lyric. If you have a beat or a groove in mind, have someone else play it over and over in the background while you riff over it. Explore the possibilities in a free-form fashion until you start to feel something that works.
Record your initial progress. Great ideas are fleeting, so when you have something that you can keep, even if it is just a couple seconds long, either write it down or get it on tape. You can even call your voicemail and record it. Don't let it slip away from you!
Expand upon what you have. Start developing your idea. If it is a chorus or a verse, think of an intro to lead into it. If it is a lyric or a thought, continue from that thought. Think about what was working during a jam or improvisation and try to recreate it.
Keep generating ideas. Go back to Step 1 and start thinking of lyrics on the same theme, riffs or melodies in the same key. Don't worry about where these new parts will fit yet.
Put it all together. Once you have all the essential parts of a song-a beat, a chord progression, a melody, lyrics (verses, refrain)-start putting them together.
Play it through. Play, sing or record all the parts in sequence. Take notice of rough parts or awkward transitions.
Put on the finishing touches. Smooth out any problem areas in the song, add vocal harmonies or extra flourishes.