That won't stop the bastards trying to claim you do not have rights and putting adverts on you Youtube videos of course. I've lost count of the times I've had to challenge them. Of course you never see any of the revenue they get from those adverts as compensation for wasting your time.
A little word of warning Jonathan. Although that youtube link is a good start, I'm not entirely convinced it's tells you the whole story.
It's my understanding that you actually need two licenses here in the UK. One from the writer, publisher or license holder of the piece, the second from the recording artist. This means that even if you used something by Mozart, that is in theory is out of copyright, you may still have to compensate the orchestra who recorded the version you wish to use.
Then of course you have the potential nightmare of dealing with the artistes. It's not unheard of for bands to ask to see scripts or rough cuts of the finished film, for band approval, before any agreement can be reached.
As well as PRS (already mentioned) check out the FAQ's on the MPA (music publishers association) website.
You need to read up on sync (synchronisation) rights. It's a complicated area for the novice and it's worth getting to grips with the various terms which the collecting societies may quote in their licences, to make sure you only pay for the licence you need and no more. If you want to use a specific song or track, then go down the PRS for Music route as that encompasses both the PRS and MCPS sides. However if you only need some generic music to fit a particular mood or event, consider getting what is known as production music (a sort of off-the-peg music) from a specialist source such as SoundCloud, who provide a simple one-stop no-nonsense licence, and tend to be cheaper than trying to deal with either the big collecting societies or the major labels.
I suspect Razoir is right. It used to be the MCPS, "mechanical copyright protection society" but may be they don't exist any more!!
Of course anything over 50 years old and the original is out of copyright in the UK, different in the US where I think it is 100 years, but I guess you are thinking of using something a little more contempory!
Whatever music you use there are two licences that you need to obtain for each piece:
Publishing licence: from the people who own the copyright to the piece of music i.e. lyrics and composition.
Recording licence: from the people who performed the version of music that you want to use. For example, sound recording copyright exists in a pianist’s recording of one of Beethoven's sonatas.
Old Master Copyright in music in the UK lasts for 70 years after the death of the composer... but don't forget the musicains, etc, also have to be dead 70 years. So you'd need a very old recording of that sonata before you could use it for free. Perhaps a 78rpm stashed away under your old gramophone.