Registering domains is done via a domain registrar. You can check the availability of the domain at the registrar or using a WHOIS service such as whois.com, which provides registrant information or confirmation if the domain is already taken/registered.
Here are some examples of well-known registrars.
Registering a freely available domain name is often referred to as hand registering or ‘hand regging’ domains.
The price can vary by registrar and TLD, when registering, transferring (moving a domain to another registrar) and renewing (extending the domain registration period) domains. Some domain registrars may offer cheap initial registration costs, but high renewal or transfer costs, so be sure to check the small print.
When registering country specific ccTLDs such as .de, .uk etc. it is often cheaper to go to a registrar that specializes in that particular ccTLD but be aware of any residency requirements that may apply. e.g. to register .eu domains you must be an organization that is established in the EU or be a resident or citizen of a EU member state.
Be sure to read which domains to select here first, to avoid wasting money.