All club teams (such as Manchester United, Spurs, Arsenal etc) play association football in the football league system.
The league has many teirs, the Premier league being the top division: http://en.wikipedia.org/wiki/English_football_league_system
It is national sides such as Brazil, Argentina, France etc who play in the FIFA World Cup and just European national sides who play in the European Football Championships.
There are various European club competions that club teams play in and there are different ways of qualifying - either by winning their respective domestic leagues or cup competitions or by finishing in a certain league position.
You'll soon get the hang of it. If you plan to watch football in London, don't bother going to Stamford Beach as it costs an arm and a leg (as does Arsenal) - why don't you watch a decent team and hook up with that Spurs fan that hangs about on here!
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