Few people have their webspace on their own computers at home, as it can create security problems, high costs and requires some technical expertise. Thats something else I should mention. I always try the free way first and then work up from there. Certain web homepage building communities like YahooGeoCities, also provide space for free, except with some limitations on advertising, and you usually have to display their advertisers, on the space in their site you use. Businesses may also have the option to use their own computer, or to use a specialized service which will either just provide web hosting or also totally design a web site to your needs.
Names are tough to come by.
The sooner you find one the better. Keep in mind, You will still need a web host. One more note, be sure the domain service is compatable with your web host. Some hosts need url redirection or fowarding to point your domain name to the host. Usually free hosts like AOL and Yahoo Geocities.