Oh boy, here we go. The number in brackets indicate how many of that series I have played. For this, a minimum of two games in the series:

Final Fantasy (8): Final Fantasy XII
Legend of Zelda (2): Legend of Zelda: Spirit Tracks
Pokemon (8): Pokemon X
Command & Conquer (2): Red Alert 2
Mass Effect (3): Mass Effect 2
Sonic Series (2): Sonic Advance 3
Super Mario Bros (4): Super Mario 3D Land
Ace Attorney Series (7): Professor Layton vs Ace Attorney/Phoenix Wright: Trials and Tribulations
Professor Layton Series (6): Professor Layton vs Ace Attorney/Professor Layton and the Unwound Future
Dynasty Warriors (4): Dynasty Warriors 5
Super Smash Bros. (2): Super Smash Bros. Brawl
Worms Series (3): Worms Armageddon
Mario Kart (3): Mario Kart 7
SimCity (2): SimCity 4
Grand Theft Auto (5): Grand Theft Auto IV
Fable (2): Fable Anniversary
The Sims (3): The Sims 3

There were other games which are part of a series, of which I've only played one of, such as DMC: Devil May Cry, for example. I don't have a massive series list, I generally tend to stick to my favorite series, unless I'm introduced to them.