In order to find out if a domain name is available or not you need to use the services of a domain registration company.
They offer a domain search facility where you can do as many searches as you like in order to choose the right domain for your website.
Use a Domain Search Tool to Check if a domain name is available for registration
Type in the domain name you are interested in.
It will now show you if the domain name you are interested an other variations is available.
You can now select the domain names that you wish to register and purchase them.
When you purchase a domain name you basically take it off the market so that nobody else can register it.
You can just register the domain name without getting a hosting service.
A hosting service is the service that you will need in order to publish a website to your domain. It provides you with web space, email accounts and so on.
There are instances where the domain name that you are looking to register it is already taken.
This means that the domain name is already taken and registered by someone else.
If for instance they were not using the domain name for a website you could try to approach them to buy the domain name from them but pricing can be expensive.
Here is how you can find out who owns the domain.