The Master is the piece of music you stream via Spotify, etc and its copyright is in most cases, in the ownership of the artist. When you sign with a label, you will give some percentage of your Master's Copyrights to the label (the percentage they will earn through streaming, etc.). These copyrights do not include the Composition (Songwriting, Lyrics) because that is handled separately.
In many cases, Composition and Master go hand in hand when the same person wrote and produced the record, but there are a few quite important exceptions. When a vocalist, instrument player, or another producer has also worked on your music, their share of your song will be put under composition copyrights. All the lyrics are written down in a text, and all melodies, chords, etc. are here very important to whom they got written and placed into a song.
Publishing rights differ from country to country and how much a publishing company can get, but in general publishing, copyright is very important when your music gets streamed on television, is under commercials, under ads, or gets played on the radio. Therefore, publishing companies like Universal, Sony, BMG, etc are collection royalties for you and will make sure you get paid.
Mechanical royalties compensate songwriters when the music they have worked on gets reproduced, manufactured, or rerecorded by third parties. Plus, mechanical royalties are coming alongside Public Performance Royalties. That is when your song gets played by another artist or DJ. The publishing companies will pay these royalties out to which your music is assigned.
Music rights organizations:
Companies like Gema, ASCAP, and Buma/Stemra are right holding and the right collection companies for your music. When your song is distributed to a DSP is very important to have it registered by a right holding company, because they will collect a part of your Radio royalties, Streaming royalties, or royalties made through performances of your music. They can also help you to prevent juridical problems when somebody uses your song against your will.