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Ok, there are national leagues (Italy, Spain, England, Germany, etc...), all of which have their own separate pools of teams whom compete against each other (thus a league). Now, each country has a national team which competes in friendly games and tournaments (qualifying and tournament matches) such as World Cup, Olympics (Under-21 national teams), Euro, etc...). There are two major tournaments, the UEFA cup and the UEFA Champions League. Teams from all of the European leagues compete in these tournaments, each tournament has different ways for a team to qualify (usually based on the size of the league and the team's performance the previous year). The Champions League is where the better teams play, with the ok teams playing in the UEFA cup.
As for seasons, the four major leagues (English Premier League, Serie A (Italy), the Bundesliga (Germany), and the Liga Espanol (Spanish) all play from August to May, not necessarily on the same schedules but close (games are on the weekends F-S-Sun) with tournament and national matches taking place during the week (usually Wednesdays).
National teams can call up players when necessary (and the players very rarely don't go). They do not have a regular season. Oh, and each league has its own league tournament which runs through out the season (usually only on weeks that there are not UEFA cup or UEFA Champions Leage matches).
Basically, the UEFA Champions league is like the NFL Playoffs (only much bigger), and the UEFA cup would be like if you too the "2nd rate" (a harsh way to put it) teams and had a tournament.
Oh and there are also region specific tournaments such as CONCAF.
Basically, world wide soccer is like the NFL on steroids. I hope this helped and made sense.