This may be a category that a family will want to purchase multiple games from. By no means is anyone restricted to just one of these games. There's a lot of good ones out there. My picks are games that I enjoy, as a 23 year old, playing with my family and friends who aren't necessarily gamers. All of these games play in 30 minutes or less. (Most of the time)
These aren't in any particular order, but I will pick my favorite at the end.
Word on the Street (there is a family version, I haven't played it) 2-10 players, divided into 2 teams
This is a great game for helping kids learn vocabulary as well as spelling. If you're not familiar with the game, here's a brief rundown. Your team gets 30 seconds to come up with and spell a word that fits the category. As you spell your word, you move the letter tiles closer to your side. You want to be the first team to get 8 letters off your side of the board. Categories are fairly simple, but coming up with a word that uses a lot of letter tiles, or specific tiles is a challenge. The game can be played by young and old alike. You could get a category such as "A type of cheese" Now, your first thought might be cheddar, but what if D is already off the board? Maybe you need to go with Swiss, that's a lot of S's. Or maybe you go with limburger. That uses a lot of letters, but be sure to spell it correctly. Word on the Street, a fun game for vocab and spelling, and a lot of fun with the family.
Tiki Topple 2-4 players
I've talked a lot about Tiki Topple on the blog, so there's not a ton of need to go into it yet again. Younger players may get upset when their tiki is toasted, they do tend to get attached to things easily, but it can also be a lesson in not revealing hidden information and making the best of a bad situation. In the game, players try to maneuver tikis up the board to match their hidden goal. No two goals are the same, but some are similar. Each round takes about 2-5 minutes, so this can be a nice game before bed, or anytime a quick game is needed.
Incan Gold 3-8 players
A great push-your-luck game with direct interaction between players. The treasure aspect of the game is fun for all ages. Even setbacks are only temporary in the game. There are interesting decisions, and it teaches little ones to not push too far. The game is played over 5 rounds, and each round can take anywhere from 30 seconds to maybe 10 minutes, but the average is probably right around 3-5 minutes, meaning the full game takes 15-30.
Can't Stop 2-4 players
This is an old classic from legendary designer Sid Sackson. The idea of the game is simple, roll 4 dice, make pairs, move up the board. The game is challenging because you can only take 3 numbers, and you have to take the first ones you roll. If you roll none of your numbers, you're out. Be the first person to claim 3 numbers all the way to the top and you win. It's quick and easy, there's a lot of fun interactions, egging on, calling people chicken if they stop too soon, or doing the "whomp whomp" when someone rolls poorly. Just thinking about this game brings a smile to my face. The time on this is hard to nail down. I've seen a game end on 1 turn, and I've seen games that take 30 minutes. I'd say that 90% of games are done in under 20 minutes, and probably 95% in under 15.
No Thanks! 3-5 players
I did a comprehensive review on No Thanks! which you can find here. The game can be a bit too mathy for young children, and there is also the strategy of knowing when to take something and when to pass that can be hard to master. It's a fun game, but it can be a bit hit or miss with kids. For time, you're looking at about 10 minutes to finish the game, which is great if you want to play multiple rounds.
For Sale 3-6 players
I recommend For Sale for older kids, probably in the 10+ maybe 12+. There's a lot of auctions, and a fair amount of strategy. That being said, houses are fun to mess with and competition for the space station can get really intense, especially when no one wants to be in the cardboard box. For time, this is probably a 15-25 minute game. A lot depends on how many players, how many rounds of betting, and how intense those competitions get. The 2nd half is usually pretty quick.
My pick here has actually shifted. It used to be Tiki Topple, which is still a good game, but I'd have to take Incan Gold these days. It's a little more expensive, but I've never met someone who didn't like it after playing. Kids seem to like the idea of adventuring for treasure, which gets them in. Even if they lose, they still collected something, which is a positive experience.
My runner up is Can't Stop. Sadly, the game is just too expensive for what it is right now. What I would recommend is mocking up your own board, finding some spare pieces and dice, and giving it a try before you buy it.
As a final note, I'm not the blogging expert on gaming with the family. I recommend two sites. http://fathergeek.com as well as http://growingupgamers.blogspot.com. Both are great with family recommendations, as well as interesting reads.