Here’s the ‘need to know basis’ version.
Neighbouring Rights (also known as Related Rights) are the rights related to the public performance of master recordings.
They literally sit beside, or neighbour the composition copyright of a work.
They apply to any audible contribution made on any sound recording. Performers who have contributed to the sound recording have a right to a share of this Equitable Remuneration.
A.K.A. more income.
Since 1997, performers are entitled to a 50% share of the income derived from the broadcast of the sound recording, commonly known as the “Performer’s share”.
The Performer’s share can include contributions made by artists, background singers, session musicians, producers, conductors etc.
If the performer does not qualify for this income or has not claimed it within the limitation period (which varies dependent on the territory), the royalties will flow back to the label or the maker/owner of the sound recording.
The copyright owner of the recordings (usually a record label) is entitled to the Rightsholder’s Share.
For more details, you can read the original piece of legislation for your territory. Or you can save yourself that degloving experience and call Good Neighbour instead.