◆ An anime year is divided into four seasons:
▷ Winter Season (Jan - Mar) (1st quarter)
▷ Spring Season (Apr - Jun) (2nd quarter)
▷ Summer Season (Jul - Sep) (3rd quarter)
▷ Fall Season (Oct - Dec) (4th quarter)
Significant for categorizing an anime into a season is the start date of the series. Also the majority of the episodes should run in the season.
Example: "A series begins airing at the end of December with 1 or 2 episodes when the Fall season ends. So it belongs to the following season (Winter season)"
A series often airs for a period of 1 season (1-cour, about 11-13 episodes) or 2 seasons (2-cour, about 22-26 episodes), but it can vary.
Update (2013-09-24):
An anime series can be tagged with several seasons, so sequels and follow-up OVAs (minimum number of 2 episodes) are no longer ignored.