​You will need to trim your rabbit's nails once per month, or as needed. Rabbits have 18 nails that are constantly growing. 4 on each rear foot, and 5 on each front foot, including the dewclaw on the inside of the leg. They must be kept short to prevent injury to the rabbit's nails, lessen the risk of injury to the leg bones or joints, and to prevent the owner from getting scratches. Unlike some dogs, rabbits can't keep their nails short by simply being allowed to run around. And sadly, rabbits don't use scratching posts! So the only option is to trim the nails.
This is an example of a nail that hasn't been trimmed in about three weeks:
A lot of beginners are nervous at first about trimming their new rabbit's nails. Most of the time, it is because they are worried about injuring the rabbit. Here's the thing. Your rabbit will forgive you for the odd mistake. Yes, cutting a bit of the quick is not pleasant for the rabbit, but it isn't the end of the world. It heals quickly and the rabbit will forget about it quickly, provided you don't freak out and make a big deal about it. With time, cutting the nails will get easier and easier until you don't even worry at all.

You can buy nail trimmers from any pet store, and will soon find what type works for you. Do not use dog nail trimmers, as they are far too big, most cat nail trimmers work fine, provided the trimmers are small. You can get similar trimmers to mine from pretty much any pet store, including Petco and Petsmart. This is the sort of thing you want:

So how do you actually trim the nails? The very first thing you'll need to do is get your rabbit comfortable being flipped over on their back. All my Hollands are flipped over frequently from birth, so they are really used to it and don't care. They actually love the attention! Some rabbits that haven't had this experience may be a bit squirmy at first. Just be patient and don't take 'no' for an answer. If you are having difficulty teaching your rabbit to hold still in this position, have an expert help you in person.

Turn your rabbit over on his back, and hold him gently but firmly on your lap. You may need a helper when you're first learning. You could even split the work and have your helper hold the rabbit while you trim the nails, or vice versa. Whatever works best for you and makes you feel confident!

​This is the position you want to have your rabbit in. She is totally relaxed and happy because she is used to being handled frequently.
On each nail there is a red or darker part called the quick. This is a blood vessel. Do not cut the quick, or you'll make your rabbit bleed. Always cut in small increments in case you cut the quick. If you accidentally cut the quick, apply a generous helping of flour or corn starch to the nail to stop the bleeding. Your rabbit will probably squirm quite a bit if the nail is cut, but it doesn't hurt as badly as they make it out. They're mostly just startled, and as prey animals, they react on instinct. Just don't panic. If you freak out, the rabbit will think there's a reason to freak out, and he or she will get scared.
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This is the sort of angle you want to trim at - exactly head-on with the nail:
Rabbits with light nails, such as brokens and white varieties, are usually the easiest to trim as the quick shows up really easily. If your rabbit’s nails are dark, the quick is a bit harder to see. You can dip them in a bit of water and hold the rabbit in bright light to see the quick better. As time goes on, you'll be able to estimate where the quick is. With mine, I don't have to do anything special to see the quick, and I just about never have a nail bleed.

You want to trim the nails fairly frequently to keep them nice and short. If you leave the nails for several months without trimming, the quick will actually grow longer and it will be impossible for the rabbit to ever have really short nails. Long nails put unnecessary stress on their joints. Make sure the dew claw in particular is always very short so it doesn't dig into the foot.
This is what the dew claw looks like: