Blacklight Retribution is one of the best free to play shooters. Runs well, good graphics, good balancing, not much 'pay to win', many modes, high population, and they've done a pretty good job of tearing out the annoyances that plague lower quality shooters.
Warframe is also good. Squad based, action based cooperative missions.
Tribes Ascend is pretty good if you're looking for a change of pace. It's draw is that it's faster paced and everyone has jetpacks. So there's aerial dodging and stuff.
However, none of those are really "mmos". They're squad based stuff like you'd get on a console.
Planetside 2 is a real mmo shooter. Open world, dynamic battles between hundreds of players for control of a planet. It's supposed to get a console release at some point as well though.
As for mmos that are not shooters, Guild Wars 2 is one of the best. But it's not totally free; you buy the game for like $40 and then you can play it with no subscription.
There's also Path of Exile. Not a real mmo since it's all instanced, but still a good free to play RPG.