One of the best ways to see if the domain name you are thinking about is available is by using a domain name lookup service. Sure you could lookup domain names by typing them into your search browser but that doesn't guarantee that a domain name is available. You can do a domain reverse lookup as well. Domain reverse lookup is basically searching for a domain with the IP address. All domains have an IP address that is your literal computer address. When someone types in your website it is converted to an IP address or points to an IP address. This is how your site is found by the computer.
There are a lot of sites to help you with this search but the following site is excellent and free http://www.dnsstuff.com/. It has all the choices you could want for DNS or IP searches. Domain reverse lookup has become a more common practice of late, as Google and other search engines will now automatically reverse lookup many search items such as phone numbers and domain names.
In the context of IP and DNS a reverse lookup is also referred to as reverse resolving, or specifically reverse DNS lookups. DNS is used to determine what IP address is associated with what domain name. So if all you have is the IP address you can still find your domain name. Lookup services vary over the Internet but the one mentioned about is a good starting point.