League of Legends - Free - Multiplayer Online Battle Arena - PC
Probably the most popular PC game at this point in time, almost all gamers and many non-gamers have heard of it. 2 teams of 5 (or 2 teams of 3, depending on which map you're playing on), and you face off against opponents in various roles. You can choose from over 100 "Champions," or playable characters, and there is a (relatively) good Ranked system.
"Champions" each have purchasable Skins, which are ways to customize how the character appears. They have varying prices, and many of them have really awesome effects.
The game is constantly being updated and changed, making it worthwhile to continue playing. There is an extensive community on Twitch, YouTube, and a fairly good Friend system allows you to keep track of your pals and invite them to games. There are also custom games that you can set up between you and your friends, and a Bot system in case you're trying to get used to the game.
You're generally matched up against people around your level, and if you're playing with friends above/below your level then Matchmaking will automatically adjust things.
You start at the beginning level (1?) and then climb your way up to 30. Once you hit 30, you can play Ranked. If you a few games each day (20-45 minutes average) you can get to 30 within 2-4 weeks.
Oh, and the developers (Riot) are absolutely fantastic. Very community-based, and they listen to what the players want. Patches (updates) come out fairly regularly, and they make sure that the game is overall fairly balanced while still offering great ways for people to stand out and prove their merit.
Personally? 9/10.
Counter Strike: Global Offensive - $15 - First Person Shooter - PC
I'm pretty sure that on the PC, it's more popular than Call of Duty. With a variety of maps, a very lucrative Skin system as well as a market based around the purchasing and selling of Skins, this game is very enticing for many people. A large variety of weapons caters to the tastes and skills of different play-styles, and a few different gamemodes as well as Competitive versus Casual offers a way to play seriously or just relax and play with friends.
(I don't really play it much, but from my friends I can tell you that it's a great game and at least 8.5/10 for most enthusiasts.)
I would describe DotA and Hearthstone, but my crush is asking to talk to me, and they take priority over writing. Peace out~