A clean drive chain accomplishes several goals. The two most worthy?
1. It makes your bike faster and will improve shifting.
2. It reduces wear on your components. Worn components are expensive to replace and will affect your shifting.
It will also help to avoid those "chain tattoos" which are annoying when applied to skin. They are downright maddening when you get them on your good work clothes as you load your bike before work in the morning.
So, it's really really dirty because you haven't cleaned it in forever or maybe never.
Here's a great link showing how to use a chain cleaner tool. You can pick one up at Dragon Sports ...we've also seen some cheap models at Walmart that get the job done.
Chain Cleaner Link
If it has had a good cleaning or didn't get that dirty in the first place, you can maintain it to avoid getting black sludgy slow gunk all over your chain, cassette, crank set and derailleurs. Just spray some lube on it after most of your rides and then wipe the chain down with a lint free rag. Wipe the crank set and the derailleur pulleys at least once a month as well.
While you are at it, you can wipe down your shifters and brake levers with a damp rag to remove sweat. The finish will tend to become very unattractive from sweat long before they actually wear out.
We'll see you and your clean speedy steed on the road.