A company that collects and distributes performance rights royalties for its members (which are copyright owners like e.g. writers or lyricists).
A sync license includes the right to use the song in a timed relation with a film or commercial. The performance rights are for the actual performance (i.e. “showing”) the respective film in the licensed media. Royalties related to the performance rights are charged for showing the film on e.g. TV. The royalties are collected by the respective local performance rights organisations (such as ASCAP, BMI or SESAC in the US, GEMA in Germany etc.) and distributed to the rights holders of the song. The amount of royalties depend on how, where and how often the film is being shown (the music is being performed).
Refers to music that is already existing before the creation of the film (in contract to composed music that is being created for the film).
A song is in public domain if is no longer owned by a copyright holder or publisher. It applies to a lot of classical songs as well as old folk songs or traditionals. A song enters public domain a certain period after the death of the writer, This depends on the respective country as well as the time the composition was created. If a song is public domain, you can use the copyright without further permissions or fees (but you need to license the respective recording). Beware: it requires a proper research and investigation to find out if a certain composition is public domain. Also, variations of an originally public domain composition (that are e.g. commonly used lyric changes) are often not in public domain. You should never assume public domain without having it properly researched.