Part of the confusion may come from the fact that British and US tv use the words "season" and "series" differently.
In the US, a tv show is also called a tv series. Each year that the show/series is on air, it has a new season. So, The Big Bang Theory, a tv series, first aired in 2007. Another way to say this is that the first season of TBBT was in 2007, the second season started in 2008, and so on.
For British tv, as least the shows from the BBC that we get here in the US, the word "series" seems to be used in the way the US uses the word "season." So the 4th "series" of Downton Abbey in the UK becomes the 4th "season" of Downton Abbey in the US.
In the US, regular tv series usually have 20-22 episodes a year. These are aired twice during the year. Frequently, a network will show the first 10 or 11 episodes, then re-run them, then show the last 10-11 episodes and re-run them. This month, a lot of series are showing their final episodes of the "TV year."
Over the summer, there will either be more re-runs or special summer replacement series, which have fewer individual episodes than the standard series that start in September/October. Under the Dome is a summer replacement series that did fairly well last summer and is coming back for a second summer season of 13 episodes. I think Buffy the Vampire Slayer started as a summer series but made it to full-time status.
And many of the cable networks, as opposed to the big broadcast networks (ABC, NBC, CBS, etc.) will start shows at various times during the year. There's probably a pattern to when they start a new season or a new series, but I haven't figured it out yet.