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[Moridin]

So, given your criteria, I'd say that here would be my recommendations. These are all games that I personally own, so I can vouch for them as good choices. Likewise, these are all games that I have taught my girlfriend how to play, and she's mostly just a casual gamer. If she can get it, so can you and your friends. Some of them are borderline, which I call out in my recommendations.

1. Chrononauts: This one is fun, semi-educational, and short. It's a simply Looney Labs game with a fun theme, and it's short--a long game would be 30-45 minutes. I think this one is superior to Fluxx (below) because it requires some actual strategy and thinking, and it requires you to interact with the choices of your opponents.
2. Fluxx: Light like Chrononauts, this one is zany and occasionally ridiculous. It's fun, game last from 5-20 minutes, but there's not a ton of strategy here. Mostly, it's a timekiller, but I find it more fun than, say, Uno. This one is best with at least 3 people, because with 2 it will get a bit boring.
3. Guillotine: This one's pretty fun, simple to teach, and fast. There can be an antagonistic element to it as you're directly competing with others over cards, but it's pretty easy on the learning curve, and, like the above two games, available for under $30.
4. Pandemic: Probably my favorite board game in existence at the moment, though there are others nibbling at its heels. It's 100% cooperative, it's a fast-play game (1 hour is a long game), it's VERY challenging but also very easy to learn (all of the game rules are more or less printed on the game board, or the cards). This is a more complex game, and you'll lose a lot when you play, but that's OK, because even when you lose you haven't spent much time with it. I also think it's an absolutely brilliantly designed game that teaches you a lot about cooperation and how different systems interact with each other. Can't recommend this one enough.
5. Shadows Over Camelot: This is one of my "iffy" picks, for several reasons. It's not cheap (probably $50), it's a more complex game than the others, and a game is going to run you and hour and a half to two hours. That having been said, this is a fantastic cooperative game, and it's the game that opened my eyes to board games as a whole. A great theme, solid components, a cooperative play that also induces a decent amount of paranoia among the players, I can't recommend this one enough if you're willing to invest the time/money into the game. This one works best with 4 players or more.
6. Small World: Another one I was iffy on based on your criteria, mostly for length of game (about an hour to an hour and a half) and cost (another $50) game. That said, this is a great competitive game that is simple to learn but very, very replayable. In fact, it's possible that this is one of the most enjoying replayable games I own, because every game is going to be different, both in outcome and in flavor. It's a competitive game, but there's also the opportunity to sort of do your own thing in the game. This one is fighting Pandemic for the top spot on my games played. Another awesome thing about the game is that it's for 2-5 players, and each number of players has a board specifically designed for that number.
7. Ticket to Ride: Another $50 game, TtR is a great game with pretty simple mechanics. It's semi-competitive, but it's a train-building game that I've taught tons of non-gamers how to play. It's a light game on the surface, but there's a lot of strategy in it. If you pick up this one, pick up the Nordic Countries version. It's specifically designed for 2-3 players, and if you like it you can pick up the Switzerland map, which is also designed for 2-3 players. This is one my girlfriend and I play a lot, because often it's just the two of us trying to kill time at night.
8. Wings of War: This is a fun game of dogfighting that absolutely could not be simpler and faster. It's also not that expensive, but it does require a nice, flat table surface to play as your placement of your airplane on the table really matters. A fun diversion game, with lots of expansions available.