A flash meter can help you with flash photography, especially when using multiple strobes. You can do the same thing by making test shots and chimping. The meter is a lot more convenient if you have to work fast, or do this a lot.
An incident light meter measures the light actually falling on the object, but it is pretty much exactly equivalent to setting an 18% grey card in the ambient light, pointing your DSLR at that, and reading the meter in it. The incident meter does this job a lot more conveniently if you have to work fast, or you do this a lot.
A spot meter can measure very small areas of the scene, which is useful for some styles of shooting (see: Zone System) and may offer more precision for this sort of thing than your DSLR can, but it's only moderately more precise and this is a pretty esoteric world. If you don't know that you need a spot meter, you're not in it (yet).
A DSLR can only measure reflected light, and even if you use a DSLR's spot metering mode the area metered can be pretty large.
Hand held light meters can not only measure reflected light, they can also measure incident light and strobe (flash) light.
They can also measure the light exactly where you want the measurement to be made.
I agree with what has been said, however there are instances when a hand-held meter will not be of any use. Perhaps not to everyone but it's a situation I run into frequently.
I shoot birds and wildlife with a long lens frequently. Measuring the light where I'm standing is meaningless since most of the time the light where my subject is (frequently up in a tree or half hidden in foliage) is completely different. Likewise, walking over where the subject is and taking a reading is out of the question since the subject would then head for the next county. The only useful reading is the one I get through my camera's lens.
Huh? I didn't think about my dslr reading the reflective light from the subject, and the light meter reading light from the source. I thought the light meter is far better than my dslr, because the light meter are crazy expensive.
Your in-camera meter will be wrong when metering subjects with poor reflectivity (e.g. someone wearing black clothes) or metering something white (e.g. a winter snow scene.) While those are just two examples, the point is your camera's meter was designed to make assumptions of average reflectivity of a subject. Anytime the reflectivity of your subject is far enough outside the norms, the meter will be wrong.
As a photographer, you will eventually learn to size-up the scene and realize when the meter is likely to be wrong. It will tend to want to overexpose black subject and underexpose light subjects (because it's trying to bring things toward that middle area.)
A hand-held light meter will nail the exposure. That's because it's not metering the subject... it doesn't care about the reflectivity of it. It's metering the actual light landing on sensor point. The more basic meters may not include support for metering flash and incident light. High-end models are much more comprehensive.
This depends on your experience with reflective metering. Incident metering is not a question of accuracy. It's perfect every time. Incident meters aren't fooled by color or brightness. They measure the light itself, not what is reflected. You still have to understand how that works and how it works in relation to dynamic range. But by measuring the light itself falling onto the scene, it eliminates guessing and getting fooled by light and dark objects.
As you may guess, I'm a fan of the hand-held meter and use it as often as I can, including during weddings, when most people would say there simply isn't time. Guess what? There's freakin' time. If there's time for Cliff Mautner to guess at matrix, take a quick look at the LCD and adjust, as you can see him do many times, then there's also time to take a quick incident reading and get it perfect on the first try. I've seen the top pros work and oftentimes their exposures start with an educated guess and a quick adjustment. A hand-held reading can be taken in the same amount of time. It's simply a style of shooting. No one style works for everyone. There isn't time for every shot in a wedding to do hand-held. But I do a variety of metering for weddings. Last weekend I used spot, matrix and manual. Some people use only spot, and some use only manual. I just use what's best for the situation.
And of course for precise flash control in manual, a meter can't be beat. You can do manual flash without a meter, but it takes a lot more guessing and adjusting.