MMORPG? FPS? Sandbox? What are all these funny words you guys keep throwing around?

RPG
Role-playing game.  You get to pretend to be someone else with more fun skills than your own.  Like fireballs or an enormous sword.  Lots of RPGs, though not all, are set in fantasy worlds with a lot of medieval elements.  Final Fantasy is a very popular and lengthy RPG series.  Often you're in charge of a group of people whose skills you manage; for example, you'll have a warrior, a mage (magic user), a healer, etc., and you direct them all in working together against whatever enemies the game throws at you.

MMORPG
Massively-multiplayer online role-playing game, like Everquest or World of Warcraft.  These are role-playing games that you play with lots and lots of other people online.  The pros?  You get to play with whoever you want.  You can form groups and do things together and interact with all kinds of people in all kinds of fun ways.  The cons?  All those aforementioned other people.  Plus, these games tend to require a lot of time.  You can get sucked into spending more time on a MMORPG than you do in a full-time job.  To be fair, if I were allowed to fireball people who annoyed me in my job, I'd probably want to spend a lot more time there too.
 
Real-time strategy
Strategy.  You know, thinking and planning and stuff.  Strategy games let you play general.  Real-time means you play at the same time as your opponent(s).  See below.

Turn-based strategy
In a turn-based strategy game, on the other hand, you take turns.  You are allowed to make your move, or a certain number of moves, and then you wait while your opponent makes his moves. 

FPS
First-person shooter.  Call of Duty or Unreal Tournament are good examples.  Your view of the game is essentially the same as if you were doing it in real life, i.e. you don't get to see your own character.  

MMOFPS
Massively-multiplayer online first-person shooter.  Like FPS, but you get to kill a lot more people.  Or be killed by a lot more people...
 
Third-person shooter
Same idea as FPS, but you get to see yourself.  Because of the wider view, you can see a bit more of what's happening around you than you can in FPS.

Open world
In an open world game, you get to wander around the entire game at will or almost at will.  Fallout 3, Crackdown, and Just Cause 2 are good recent examples.  There may be areas that you aren't strong enough or don't have the tools to reach in the beginning.
 
Sandbox
Sandbox games let you roam around the world pretty freely and usually feature lots of people, objects, side quests, and no end of other fun.  Grand Theft Auto or Crackdown are examples of sandbox games.  

Sidescroller
Sidescrollers are 2D games like the original Super Mario Brothers or most of the Castlevania series.  

Platformer
These games involve lots of jumping from one platform to another, usually with various obstacles and enemies in your way.  Again, the original Mario Brothers games and the Castlevania series are good examples.