A domain name is the website address of your site. So, for instance, for www.google.com, the domain name is google.com. The .com or .net portion of the domain name is called the extension or the suffix. The most popular (and sought after) extension is .com. After .com, the most popular extensions are .net, .org (which is used primarily by non-profit organizations), and .biz (not recommended as it's sometimes used by scammy companies). There are many more extensions, mostly by country, but you should always try to get .com. As of 2014 new domain extensions were released for different types of websites. So instead of .com you could get, say, .builders, .cleaning, .florist, .contractors, .plumbing, .diamonds, and hundreds more. However, as of 2018, these domain extensions never caught on, so I recommend against using them.
Domain names are registered with a domain name "registrar". The registrar often handles web hosting as well, so it's usually best to use the same company for your domain name and web hosting. You can go with a company like godaddy.com or register.com. But if your web designer also handles domain names and web hosting, your best bet is to go through your web designer as they can set up your accounts for you.
A domain name costs an average of $10 to $30 per year depending on the company you use and how long you register the domain name for. A longer registration period often means a lower price per year. You can register and pre-pay for a domain name on a yearly basis, for one year, or up to ten years. It's best to go with a longer period; if you register for a shorter period and you forget to renew, you could lose the domain name and someone else can buy it.
One of the most important things to remember about registering a domain name is to make sure you are the registrant (the person who legally owns the domain name). Find out why and how in my blog titled "Domain Name Theft".
Your website's files must reside somewhere other than a personal computer, in order to appear online. So they reside on web servers, which are big, fast computers with lots of memory and space, much more than most people have available in their home or office computers. Web hosting is basically the renting of space on these servers, and the servers "host" your web files.
Web hosting is usually done through a web hosting company (like GoDaddy), which is often the same company that hosts your domain name. Hosting is usually prepaid for a year or more, up to ten years, and can cost around $150 per year. As with domain names, you often pay less per year if you prepay for a longer period of time. If you forget to renew, the web hosting company will remove your files from their server and your website will no longer appear online. However, if you have copies of your website files, once you re-purchase web hosting you can re-upload the files to those servers and your website will be live again (your web designer should keep copies of your website files).