How should I keep the vegetables from flying off the board when I cut them?

Use a sharp knife, dull knives don't slice, they split the same way an axe splits logs and that will generate that sideways force that throws carrot bits.

Also learn to practice your slicing and chopping technique. You shouldn't be dropping the blade straight down like a axe or guillotine. You should be moving the blade in a orbital movement, so that you are slicing in a forward direction while the blade is on its way down to and through the carrot. By slicing properly much less of your veg will have that energy buildup because they won't "snap" apart but will be sliced apart neatly.

Search YouTube for Basic Knife Skills and watch several videos, then practice, and start slowly. You will get faster with time and time only, don't rush it or you'll get sloppy and may hurt yourself.

Both Escoce's and Elendil's answers are great: the key is a sharp knife, and the vast majority of people (in my experience) do not have very sharp knives in their kitchens. When people come over to my kitchen and try to cut something they are often shocked at how easy it is. If your knives slide off of food (like onion skins or tomato skins) rather than cut cleanly into them, they aren't sharp enough to make slicing easy.

The one thing I would add is to start with a long knife, and learn the "rocking" chop motion first with it. That is, keep the end toward the tip of the knife on the cutting board, and only lift the back end of the knife high enough to clear the carrot. With a smaller knife, you need to do what Escoce mentions and make an "orbital" motion rather than straight down. I think that's harder to do properly for beginners, so start with a big knife and rock it until you get the hang of it.

I tend to use a 10-inch chef's knife to chop carrots; minimum of 8-inch. Lots of people see such big knives and think they are too scary to use. But they really make chopping incredibly easy, especially if they are kept sharp. With a big sharp knife, you should barely feel like you need force at all to slice quickly through a carrot.

Use a bigger, sharper knife. If you're having to apply so much pressure to cut a vegetable that it's flying off into the ether, your knife isn't sharp enough. Furthermore, a blunt knife is a dangerous knife, because it is more likely to slip off the surface of the thing you're cutting and end up in your hand.

If you are not bothered about the shape of the vegetables, you can cut the (for example) carrot along its length, then place the flat side against the chopping board. You can then slice, dice or whatever with much less effort, and the carrot pieces won't roll off the board.

I agree with the other answers (particularly using a sharp knife). When I'm cutting something that tends to fly away like carrots, I'll keep the tip of the knife touching the cutting board.

It gives you a reference touch point so that you don't quickly cut through the object and hit the cutting board if you apply pressure. It also makes you cut it at an angle similar to scissors or one of those paper trimmers/cutters that you see in offices.

It's crucial to consider the chopping surface here. If you are using a glass or ceramic board you will have problems with slippage.

Use a wooden or plastic board, and (as others have said) a sharp knife of a suitable type.

Beyond the comments about knives, which is right on, I use cafeteria trays with sides in place of a standard cutting board for the majority of my chopping and cutting. I've never had the dreaded roll off problem and I can carry the trays from place to place without fear of the same vegetable fly off.

I agree with many of the suggestions above. However, if your carrots are still rolling off the cutting board, consider putting one or more rolled towels around the edge of the cutting board to prevent the "rollers" from escaping their inevitable fate.