This depends on a lot of particulars and what kind of "label" we are talking about.
In general, you and any co-writers/lyricists, etc., will copyright the song (the complete entity of the marriage of your music and his/her/their words and melodies), to which you will both retain rights to, either as a default 50/50 split, or based on whatever agreement you come up with which should be signed in a paperwork agreement commonly known as a "split sheet."
The actual "master," or the final-final master recording with all music, vocal parts, mixed, and mastered (plus any "versions," stems, or other mixes) would be under copyright of the label. You will probably license it to them and they will administer the copyright for the sound recording itself (separate from the song/composition embodied therein). This allows them to make "records" (whether physical copies, MP3 downloads, or streamable files), for which you/they will receive royalties based on specific rates as indicated in your contract. If you are not signed, but the artist is, then you need an agreement with the artist and/or label spelling out your terms and royalty rates, and those should be paid to you before the artist gets their share of record royalties (should be in quotes; nowadays doesn't have to be a physical copy but can be).
This is separate from publishing income, which is from the copyright for the _song_, not the _recording_. Some labels have affiliated publishing companies that they expect you to sign with, others don't. Artist-owned publishing companies are pretty easy to set-up, so if you don't have one, and/or your artist doesn't have one, then you should look into that asap.
It all gets pretty complicated, I am not an attorney, and I'm not dispensing legal advice (!), so you need to do your own due diligence and research. READ the books by Donald Passman, Moses Avalon, and Kashif, as well as "This Business of Music" (forget the author's name). That is where you will source answers to specific questions. My best advice-- Do Not take what even well-meaning denizens of Internet forums post as Gospel (even at as great a place as FP, and even from me). Do your research before you sign anything!!!!
GJ
