First you need to think of a domain name you would like to use and then check to see if it is registered. Most hosting and dns companies allow you to check the availability of a domain name from their website. You can register it once you've found a domain name you like that is available.

Usually the webhost you choose can and will register your domain for you. Make sure that the company you register it with registers you as the owner. Some less than honest companies register themselves as the owner and use this to lock you into their service because they control the pointing of the domain name. Also ask about the management of the domain name. There should be a way for you to change the information in the DNS record so that you can move your website to a different host or point your email addresses to a different server. Usually the company will provide you with a web interface that you can use to change this information.

Make sure you know what your doing before you change the DNS informatin yourself. Normally you just want to "point" your domain name at the name servers of your webhost. Your webhost can tell you the address of their nameservers. It can take up to 2 days for your DNS information to "propogate" through the internet allowing everyone around the world to get the corrent information.