The songwriter/composer of a song (not a recording of a song, but the actual melody, lyrics) gets a performance royalty each time a song is played in "public" (internet and broadcast radio, in the elevator, at a bar, etc). This is the royalty BMI, ASCAP, and SESAC collect. If the song is recorded and sold by someone, the songwriter gets mechanical royalties for each unit sold. If memory serves me correctly, the compulsory rate right now is 8 cents per unit sold. If the song in any form (recorded, sung by a drunk dude, etc) is used in something like a movie or a TV show, the songwriter gets a synchronization royalty. I've seen sync royalties range from $5,000 to $250,000. Songwriters are usually signed to publishing companies. Publishing companies are mostly owned by record labels themselves (or their parent companies). Publishing companies take a cut of the songwriters' royalties, but not as big as the cut record labels take from recording artists. I've seen rates ranging from 10% to 30%.
On the other hand, the recording artist gets pretty much whatever the record label decides to give them as described in the recording contract. The label will own the song recordings, not the artist. Recording artists (and record labels) do not get any royalties for public performance. Yes, when a song is played on FM radio, the record label doesn't get a penny. The only exception to this is when the song is played on "interactive" services on the Internet. This is the royalty Sound Exchange collects. In those cases, both the songwriter and whoever owns the copyright to the sound recording (the record label) get royalties. The main sources of revenue for the label are from these royalties, and from selling the sound recording in stores and online. They take a large chunk (60-70%) of this and distribute the rest to the song's recording artist, producer, etc.