Disclaimer: This is my preferred shortlist to introduce the power of free to play. I did not mention a lot of good free to play games here as there are thousands.
These games are multiplayer and co-op only by definition, you're gonna need a decent internet connection to play in foreign servers. Few games may let you to host your own server.
Battlefield 3 was free for a week last month, those who added the game in origin can play the multiplayer forever (I hope all of you added it).
Now I share some free to play original multiplayer games-
steam
✔ Planetside 2 (well balanced sci-fi mmo fps)
✔ Warface (new fps with good graphics, shooting like cod/bf)
✔ Ghost recon phantoms (great tactical thirdperson shooter but ping problem as all servers are in eu)
✔ Warframe
✔ Backlight Retribution
✔ Warthunder (if u like dogfighting, armors and troops)
✔ Hawken (do u like mech but didnt buy titanfall? hawken is here with decent graphics)
✔ Heroes and Generals (new world war fps+strategy)
✔ Unturned (survival game, graphics like minecraft)
✔ Tribes ascend (movement too fast, may remind you of UT)
✔ Marvel Heroes
✔ Star Conflict
✔ Trackmania
last but The BEST in steam-
✔✔ TEAM FORTRESS 2 (comment not needed, the game speaks for itself)
origin
✔ Fifa world (those who wanna play fifa online but didnt buy fifa)
✔ NFS world (probably best f2p racer)
other software-
✔ World of tanks (if u like armor)
✔ World of warplanes
mmo i didnt mention- dota2,lol,SWToR, LotRO,DC universe, PoE and MANY MORE... just google for top f2p games.
also, f2p survival mmo H1Z1 is coming soon, I will be buying early release.