Barclays Premier League (henceforth BPL) starts in the month of August (after the FA Community Shield). Quoting from Wikipedia,
During the course of a season (from August to May) each club plays the others twice (a double round-robin system), once at their home stadium and once at that of their opponents, for a total of 38 games.
BPL's fixtures are intertwined with other club competitions like Capital One Cup, FA Cup, Euro League and the Champions League. Most domestic league competitions would be similarly scheduled, intertwined with similar domestic cups and such.
Champions League starts later than BPL -- in September (the group stage) and ends later than BPL -- in the end of May. Champions League Final marks the end of the European club football season.
The FIFA World Cup fits pretty well with the club season. The upcoming World Cup is scheduled from the second week of June to the second week of July and was the case for previous World Cups as well. In other words, right in between the two club football seasons.