There is no singular soccer season, with different countries observing different calendars. In the UK and central Europe, most leagues follow a 'winter season' (a season which kicks off around Fall and is played through winter into the spring months), but in some Scandinavian countries, where weather can be more severe during the winter months, they play a 'summer season' - the Swedish Allsvenskan is generally played from March until November, for example.
So in a sense, the soccer season never truly ends! However, there tends to be harmony in certain regions.
The elite European leagues, such as Premier League, Serie A and La Liga usually begin around late summer and end in late spring - around the end of May.
In the United States (US), soccer officially starts in Fall for youth soccer. District tournaments usually follow the youth divisions and school-sponsored league, through Autumn. In contrast, the Major League Soccer regular season generally starts in February and runs until October, with MLS playoffs following from October to December.