performance royalties are royalties you get for having your songs performed, ie. radio, clubs, bars.
Mecahnical royalties are royalties you get for the permission of having your song replicated. A label has to pay mechanicals in order to press up CD's. Movies and TV shows have to pay mechanicals in order to use your music. Bands have to pay mechanicals in order to cover your songs, and have them affixed to a CD.
There are:
sales royalties
synchronization royalties
publishing royalties
overseas licensing royalties
Licensing royalties
I'm not getting into explaining all of these. There are plenty of books, and web articles.