A song is owned by its publisher. Unless you're a songwriter with a successful track record like Diane Warren, publishing deals are usually done in conjunction with recording deals, and if you're lucky enough to get both kinds of deals, or a 360 deal covering your recording, publishing, touring and merchandising, you may even get an advance of money against your future songwriting income from record sales, which is known as mechanical royalties.
Another way to make money on a song is have it synchronized with the images in a film, tv show, or for an advertisement/commercial. That is known as a synch license, and usually the license fee is split 50-50 between the owner of the master recording of a song (usually the record label) and the owner/publisher of the musical composition of the song, who in turn pays 50% of their 50% to the songwriters(s).
Self-Publish, or Find a Publisher.
If you're not a recording artist with a recording deal and a publishing deal, then you can either self-publish your songs, or find a publisher.
In a publishing deal, the publisher, as owner of a song usually keeps 50% of the income, and pays the songwriter(s) the other 50%. If you're self-published, you get all 100%. If there are multiple writers, the writers' 50% is divided amoung the writers in whatever splits they've agreed to. Usually splits are equal and pro-rata, but not always, depending on the stature of a writer.
Register With a Performing Rights Society
You can register with a performing rights society as both a publisher and a writer. There are 3 perofrming rights societies in the U.S., ASCAP, BMI and SESAC. These societies collect money on behalf on their songwriter and publisher members, and pay their members directly for uses of their songs.