You don't state your ultimate goal so I shall assume that it is recreational. If you are thinking about becoming a professional pilot this answer is not for you.
I have also assumed that you want to fly conventional aeroplanes rather than microlights, motor gliders or home-built aircraft.
Until last year licences were mainly national but now a new set of European licences are available (EASA licences). Things are excessively complex at present because we are in the middle of the transition from the old to the new.
The CAA publish a totally unreadable document called CAP 804 that gives more details on all licences than you will ever need to know. (Search for it on the web.)
The basic licence private pilot's licence is the PPL(aeroplanes) or PPL(A). This is the most expensive licence to get but is recognised worldwide. Most flying training clubs will be more than happy to teach you to get this licence.
A cheaper option is the European Light Aircraft Pilots Licence for aeroplanes (LAPL(A)). It is recognised in Europe. To upgrade it to a PPL(A) involves a bit more training and a slightly stiffer medical. It is a new licence, introduced only last year, so not many training organisations are up to speed on it.
The cheapest option was the UK National PPL (NPPL) but most holders will have to change to the LAPL(A) by 2015 so unless you complete your training before April 2015 you will end up getting a LAPL(A).