UPDATE! This post is quite old. For an updated version of this post with example pictures click here.
Umbrellas throw light EVERYWHERE! Because they are large light sources, they produce very soft light. The biggest draw to umbrellas is frankly the cost, portability, and simplicity. A 43″ umbrella can be purchased for around $17.
While umbrellas are easy to use because they throw light everywhere, it is also quite limiting. Flash photographers like to be able to control the light and have the latitude to be able to throw light only on a certain area of the subject. This is practically impossible with an umbrella.
You should know that there are two basic types of umbrellas. Shoot through umbrellas are aimed at the subject and goes between the flash and the subject. This light is slightly more controllable than the other type of umbrella: the reflective umbrella. Reflective umbrellas go behind the flash. The flash fires into the umbrella, and then the reflective umbrella kicks the light back to the subject. This light is almost impossible to control.
Keep in mind that umbrellas come in many different sizes. The larger the size, the softer the light. If you're looking at buying umbrellas, I'd recommend this one from Westcott. It's built solidly and comes at an amazing price.
I like to explain soft boxes as controllable umbrellas. It basically offers the benefits of soft light like an umbrella does, but it throws a narrower beam of light. This allows photographers to hit only a part of the subject with soft light.
Another benefit to a soft box is that it produces light which is more even than light coming from an umbrella. The only thing between the bare bulb of the flash and the subject in an umbrella is a thin sheet of white material in an umbrella, so the light in the center is significantly hotter than the light around the edges of the umbrella. In a soft box, the light goes through two pieces of material which spread out the light.
For these reasons, professional photographers tend to favor soft boxes over umbrellas in many situations. I usually only use umbrellas for grab-and-go type situations where I just need light over a large area quickly. I prefer a softbox for more detailed lighting situations.
I have to laugh when I read blog posts and comments by photographers when they talk about beauty dishes. I recently read a blog post on a different website where the photographer claimed that the beauty dish produces the softest light of any light modifier. Hmmm… nope. The beauty dish is a very trendy look in photography right now, and I think the hype produces a lot of fallacies as well.
A beauty dish produces an edgy and crisp light that is favored for showing shape and texture in the subject. For example, a beauty dish is a common type of lighting for sports portraits, which typically use a crisp type of lighting to highlight muscles, curves, and texture. I certainly don't mean to say that beauty dishes are ONLY useful for sports portraits, but that this is a common place where they are used.
You can buy this beauty dish for your speedlight for just $99. I haven't tried it myself, but I've heard good reports on it. You can also check out this link for directions to make a homemade beauty dish for your flash.