You need permission to use any copyright music. You need to get two separate permissions, for the publishing and the recording. Music publishing is dealt with by PRS and music recordings are dealt with by PPL, who will give you contact details of the record company so you can deal with them directly.
If you perform your own version of someone else’s song you still have to deal with publishing rights (through PRS) if the music, words or arrangement are still in copyright.
If you use a recording of a piece of music written by Beethoven you still have to deal with recording rights (through PPL) if the recording or performers’ rights are still in copyright.
You may need to specify exactly how the music will be used (‘sync rights’) and to seek permission again if you change this (eg using it with different images).
If you create your own music using loop-based software you are normally allowed to use it without restriction, but check the terms and conditions.
The Open Music Archive has downloadable music recordings which are out of copyright.
You can also look for Creative Commons music.
Musopen.com has downloadable performances of out-of-copyright classical music, under the Creative Commons CC0 license (which doesn’t place any restriction on how you use them).