I can help you find and purchase your domain name from a Registrar. (The domain name is what you type into the address bar of your browser) Domain names are relatively inexpensive. The shortest amount of time that you can purchase a domain name is one year. It is your property for however long you decide to make it that way. Even if you decide to take down your web site or move it elsewhere, you will still own your domain name as long as your payments to the Registrar are up-to-date. This gives you the flexibility to completely change your site, your host server, put the Domain Name up for auction, or whatever you want to do with it. It is yours.
A domain needs to be hosted if it is to be a website. I no longer provide hosting services, but will set you up with your own hosting account. This enables me to work on your site and monitor as requested. You will own your domain and host server and will be responsible for renewing your site as desired.