Well, taking 2003-2013 into consideration is hard. There were quite a few great series during this last decade, but I can't think of one series that had some spectacular games that also didn't have some truly terrible ones.
With that said, I think I'm going with Call of Duty.
Yes, I said it: Call of Duty. Between Call of Duty, Call of Duty 2, Call of Duty 4: Modern Warfare, Call of Duty: World at War, and Modern Warfare 2, I can't think of a single series that delivered such a high quality experience over the course of the decade. Yes, the franchise has sort of been complete crap ever since IW abandoned improving MW2 in favor of making more money off DLC. Yes, I did comment that Black Ops was possibly the worst game I ever played, but that sort of comes with the territory of being the worst game in such a great franchise and the one that sort of sealed the deal on its fall from glory. However, the first two Call of Duty games are my #2 and #3 WWII shooters respectively (only losing to Allied Assault, which is actually the spiritual predecessor of Call of Duty). CoD4 is still the best online FPS I've ever played, and MW2 is not far behind. Even CoD3's campaign was decent, and World at War was an incredible shooter in its own right, even if it was just a WWII mod for CoD4. Overall, regardless of its problems of late, Call of Duty has simply provided the best memories of this generation for me (so long as we ignore stand-alone games).
Other series that come close include:
Prince of Persia--This would be my choice, but the 2008 game and The Forgotten Sands prevented that. I would have said it was the greatest if the worst in the series was Warrior Within. Sadly, that wasn't the case.
The Elder Scrolls--This is from my love for Oblivion, mostly. But when I consider my apathy towards Morrowind and completely negative feelings towards Skyrim, I just cannot say this.
Fallout--Again, mostly because of Fallout 3. Again, though, that is only one game from the series. New Vegas just fills me with apathy.
Mass Effect--Next to Prince of Persia, this came the closest. However, my feelings towards the first game are just too negative, and my feelings towards the third game range from apathy to negativity far too often, even if it was a good game on the whole. Mass Effect 2 is really the only game in the series that competes with the better games in the CoD series.
Other than that, I don't consider "series" like Galaxy to be actual series (otherwise, I'd have put The Sands of Time trilogy on top), just mini-series within their respective franchises.