Nylon is an inexpensive and popular rope. It's smooth and shiny, and is easy to find. On the down side, it doesn't hold knots very well. Also if it runs across your skin very quickly it can give a rope burn faster than cotton rope.
Cotton is also inexpensive, stretches a little bit, and holds knots very well. It's not the strongest rope though, so you shouldn't use it for things like suspension or where there's a lot of pressure on the rope. Watch out for cotton clothesline that has a hard nylon core. It can jab you.
Opinions vary on the cheap cotton rope you can buy at places like Home Depot. Some people insist that it's junk and will fall apart after a couple washings. I'm kind of on the fence about it. It's certainly not a high quality rope, though I've used it many times without much problem. If you can afford to spend more though, I'd definitely recommend hemp or a high quality cotton rope.
Hemp rope is very nice and is often associated with Japanese bondage. The rope has to be heavily treated though before it's useful for bondage. Twisted Monk is a great source for pre-treated hemp. Hemp is very expensive, but it's wonderful stuff. It handles great, makes a great sound that I always enjoy when it uncoils and drops to the floor, and it even smells nice.
Diameter - Rope for bondage is usually 1/4" to 3/8" inch thick. A larger diameter can be hard to work with, though some people do prefer to use heavier ropes.
Finishing the ends - There are several ways to finish the ends. I like to just wrap them tightly with electrical tape. Sewing or "whipping" the ends looks nice, though it's very labor intensive. Nylon rope can be melted over a flame, but be careful - the molten blobs it produces are extremely hot, and after cooling they can be sharp and scratchy. I gave up on this technique after ruining too many pairs of pantyhose. I've also heard of people dipping the ends of rope into latex or "tool dip" that you can find at the hardware store.
Washing - rope gets dirty over time, especially if you're having sex with bondage. I like to use one of those lingerie bags to keep the rope from getting all wrapped up and just throw it in the washing machine with a mild detergent and then let it air dry, hanging it in large loops. Cotton will take a few days to dry completely.
I just use common sense in terms of how often to wash it. If you're just tying up people fully clothed, you can probably go a long time. In her Japanese bondage book, Midori makes the suggestion that you can store dirty rope away from your clean rope, and identify it by coiling it up in a different way, for example, making a daisy chain with it.
Although some people love to do bondage with one long piece of rope (100 or 200 feet or even longer), I'm more comfortable using short pieces. There is no optimal length that your rope should be. Basically, whatever you need is the right length.
I have, however, found useful Jay Wiseman's recommendation of using rope cut into lengths that are multiples of 6' feet: 12, 18, 24, and 30 foot lengths.
Twisted Monk sells his hemp rope at lengths of 15, 30 and 50 feet. The 15 and 30 foot lengths always feel like good lengths to me.
Your ropes will probably get shorter over time too, as you cut off sections that are damaged or unraveling.
Rainbow Rope - Many different kinds of rope at good prices
Twisted Monk - Excellent conditioned hemp rope, a bit pricey but worth it
Like I mention above, I've also bought cheaper, less quality rope in the past, even stuff from the .99 Cent Store. Whatever you're most comfortable using is probably the right rope for you.