Generally, you always start by picking out a root note, chord or chords, a scale, that sort of thing.
Kick & Bass method: This is when you create a drum loop and bassline, and let your rhythm guide the rest of the composition. Good for techno or even jazzy stuff, or really anything with an emphasis on drums or speed/high tempo.
Phrase method: This is when you create an arpeggio or short melody that you base at least the very first part of your song or intro around. This is probably one of the easier methods, at least for me. This method is useful because creating an arpeggio or phrase helps you establish both the harmony and the rhythm at the same time. It's good for improvising a song, and it's good for folk-styled music.
Melody method: This is when you create a melody first. This can be tricky to harmonize, as it requires working backwards after finishing the melody and figuring out what chords (and sometimes meter) the melody is in. This is good for emotional stuff, solos, etc., and if you play your music on a piano, or you're doing something with a sparse harmony (like a flute + harp), you generally shouldn't need to work backwards.
And lastly:
Chord method: This is when you choose a meter and then decide the chord progression for the whole song or part of the song, all before doing anything like basslines or drumloops or melodies, etc. This is probably the best in terms of musical form.
Chord method is sort of like how a sketch artist starts with the circles and lines to create a skeleton before adding all the details.
You might be surprised by how many rhythms and melodies and phrases just come to you when you have your chords in front of you and you know exactly where your song is going.
Chord method's good for pop music, bubblegum sort of stuff, or anything you want to have a very specific sound and you already know how to get that sound, like I would use this method if I wanted to do a medieval sounding song because this style is very formulaic and involves using a picardy chord at the end of your progression.
Chord method has its limits, though. Sometimes I want an unrefined or more chaotic sound, and that's when I use the Phrase method.
I also find the chord method to be the least fun.
Edit: Man, so many typos.
Edit 2: Woo. I'm tired tonight.
Edit 3: Good thread necro, guys.
I wish the dates on posts were bigger so I would be more inclined to look at them.