Put simply, the performing right is a bundle of rights that copyright owners have in their songs and lyrics (or 'works' in legal-speak).
It includes two of the Restricted Acts that are described in 'How to... Copyright' - the right to perform a work in public and the right to broadcast it.
In practice, it covers a whole raft of activities from playing a song live at a gig or radio session; playing a recording of a piece of music as part of a radio or TV broadcast; or playing music in a club, pub, shop - even an aerobics class.
Anyone wishing to use copyright music in this way needs a licence from the owner of the work. The performing right is what's known as an economic right - the owner of the work can charge a royalty for granting the licence.
Potentially, it even includes you performing a tune you've written with your own band. For example, whenever Oasis tour, Noel Gallagher gets royalties as the writer of their songs. The fact he was also part of the performance makes no difference.
Likewise, the other performers in a band will not receive any performing right royalties unless they are also credited as writers of the songs.