What's the best board games that 1) can play with 10-yr-olds and 2) can finish game in 30-60 minutes?
A lot really depends upon what type of game(s) you're looking for and that you think your family will enjoy. Some games are equally great as a family game or with a group of adults. Some games are cooperative, some are more or less player interactive, some have strong theme elements that add to the enjoyment depending upon what you and your kids like. Ticket to Ride will definitely push the time envelope but is a feast for the eyes, simple to learn/play, and equally entertaining for adults and families. It's one of the games I introduced early on to my kids. Niagara is another beautiful game that isn't quite as universally appealing as Ticket, but kids will get a real kick out of watching canoes roll down the river and descend over Niagara Falls (uses clear discs to simulate the river inlaid into a 3-D contoured board laid atop the box top and bottom to simulate rapids and allows canoes to travel downstream and often very literally take the ultimate plunge).
However, some other games that have very entertaining themes include King of Tokyo (giant godzilla type monsters running amok and fighting to become king of the hill Tokyo ), Forbidden island (cooperative game where you are racing against time and working together to find all of the treasure and escape from a sinking island), Fearsome Floors where you are trying to play hide and seek and escape from a scary monster who will eat your people. These are all games with VERY strong thematic elements that may appeal to your kids.
Here are several fun family games for your age group + anticipated playing time (although until you really get to know a game - double the time)
King of Tokyo - 30 minutes (especially if you have boys is a fun and fast game where the monster battle theme is a guaranteed winner)
Loot - 20 minutes (only about $10 fast fun easy to play card game with a cool pirate theme)
Hey That's My Fish - 20 minutes (might be a bit too youngish for 10 year olds, but very easy to learn abstract strategy game)
Ticket to Ride - 45 to 60 minutes (universal appeal and is a great gateway game for kids and adults alike)
Fearsome Floors - 60 minutes (Tense game and if the kids are going to get upset/scared with monster chasing and eating their people - might want to avoid)
Niagara - 45 minutes (beautiful game kids will like as much for aesthetic value, but I've found to be polarizing with adults)
Around the World in 80 Days - 60 minutes (Traditional move around the board start to finish type game with enough bells and whistles to be fun for adults too)
Bohnanza - 45 minutes (card game where you are constantly interacting trading cards - NO DOWN TIME - which is really cool and keeps everyone involved)
Manhattan- 45 minutes (cool 3d building skyscrapers with your building blocks but more than enough strategy to entertain adults as well as kids)
Saboteur - 30 minutes (fast and fun cooperative/traitor card game where you lay out a tunnel path to reach the gold)
Forbidden Island - 30 minutes (100% cooperative, very easy, very tense, hard to beat the game, but this theme shines especially with girls)
Castle Panic - 60 minutes (100% cooperative, hard to beat the game, kill D & D type monsters before they overrun your castle, better theme for boys and those of us who like to kill orcs, trolls, etc.)
If time is the biggest factor for you, then I recommend Loot, Saboteur, King of Tokyo, and Forbidden Island.
If you get Ticket to Ride, be sure to get the 1910 Expansion which provides full-sized cards and increases the number of routes you can build from 30 to 69. Best expansion to a game Evah (but it's so essential that it ought to come with the game in the first place IMO).
This is by no means a comprehensive list and there are tons of games I've left out that are also great family games, like Carcassone, Settlers of Cataan, and a few others that have a HUGE following. I recommend checking out boardgamegeek.com before you make a purchase, as you can find reviews, videos, and lots of other details that will help you in making the best decision(s).
Good Luck!