Performing rights royalties are royalties paid to a songwriter whenever one of their songs is performed. If the songwriter is a member of a performing rights collection agency, like BMI, ASCAP, or PRS, the collection group will keep track of performances of the songwriter's material, collect payment, and pass the royalties on to the songwriter. Performing rights royalties are paid on radio plays of songs and songs played in clubs, in restaurants, in bars - anywhere music is played. The royalties are also collected whenever anyone does a cover version of your song.
It is important to note the distinction between performing rights royalties, which are paid to the songwriter exclusively, and mechanical royalties, which are paid to all of the musicians involved in a track. This does mean that songwriters in a band stand to make more money than the other people in the band. For instance, if your band plays a show at a venue that is registered with a performing rights collection agency, that venue will report the songs you have played to the agency, who will collect royalties from the venue and pay them to the songwriter in the band.