Develop an idea. It may be something you consciously work out, or a song title, lyric line or melody you can't stop humming. Whatever generates the initial idea, get it down on paper in rough form before you lose it. If you want to compose songs regularly, make a habit of carrying around a pencil and notebook wherever you go. Jot down lyric ideas, ideas for song titles and even describe little musical phrases you hear in your head.
Sit down in a quiet place with the composing instrument of your choice. Many songs are composed on acoustic guitar or piano. Even if you aren't a guitar player, consider learning basic chords and how to play them. This is an easy, effective way to capture the essence of a song idea until you can work out the details later on. A lot of singers and songwriters who aren't skilled musicians will learn the basics of guitar for this reason.
Use plain notebook paper to write down your song lyrics. Strum chords over the lyrics to get an idea about the key you will play in and how the chords will change over the lyrics. As you find the chords for your song, write them above the lyrics where the chord changes occur. This will give you a lead sheet for your song--something you can work with to get details. A lead sheet is nothing more than chords and lyrics. This will be your starting point.
Practice from your lead sheet. Try strumming the chord changes in various rhythm patterns until you find one that feels right to you. Songwriting is a personal thing. Nobody can tell you you're playing the right or wrong way when you're writing a song. You'll know when you have the rhythm of the song the way you want it, when the chords sound good to you over what you're singing and if you like the melody you're singing with the words. You may not get it right away, but continue to play until you begin repeating patterns and the song gels. Once this happens, record what you have and listen to it for basic structure.
Add details to your basic song structure. How you do this will depend upon your level of musicianship. If you can read and write traditional music notation, get some blank staff paper and write out some of the musical embellishments that will fill your song out. If you can write parts for all of the instruments, feel free to do it yourself, but if not, you can find friends that play and collaborate with them on their parts. A lot of band members tend to write this way. One or two songwriters will come up with the chord changes and lyrics, which is the main body of the song, and the band members will write their solo parts.
Record your song with a band. Be sure to allow room for the song to grow on its own during the recording process; this happens frequently. A song is not set in stone until it is recorded. During the process, especially if you are open to collaboration, the song you initially heard may grow into something else as different musicians add their musical personalities. This is the magic of good songwriting.