What is a domain name?
It is the address people type. It is not the website, it is not your email, and you rent it rather than own it.
A real Clay site, built from a Google listing
You are probably reading this because somebody quoted you for a website and the quote had a line on it you did not recognise.
A domain name is the address, and that is genuinely all it is: yourshop.com is a name that points at a computer somewhere. The website is what is on that computer, the email is a separate service that can use the same name, and the yearly fee is rent rather than a purchase.
The four things worth knowing
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1
It is rented, one year at a time
You register it for a year or several and renew it. Miss the renewal and it goes back on the market, which is how businesses lose an address that is printed on their van.
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2
The website is separate from the name
You can move the site to a different builder and keep the address, or keep the site and change the name. They are two things, and a supplier who bundles them can make that hard to see.
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3
Email can use it, and is another thing again
hello@yourshop.com is a mailbox service that happens to use your domain. It is not included by owning the name and it is usually a few euros a month.
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4
It has to be in YOUR name
The single most expensive mistake in this whole subject: a domain registered in the name of the person who built your site is a domain you cannot take with you.
The bit after the dot matters less than you think
A country extension is fine.com is fine, and the newer ones are fine. The one real consideration is whether somebody can hear it and type it, which is a spelling question rather than a status one.
Where the extension does matter is trust in some markets, and that is local knowledge rather than a rule: look at what the businesses near you use.

A real address exists before you buy anything
A Clay site is live at name.clay.site for nothing, and that is a real address anybody can open, print or send. Plenty of businesses run on it for a year and buy a name when they are sure of the name.
And buying one later costs nothing in lost work: the free address keeps working, so a card or a sticker printed today does not become wrong.

What a domain is not
What it costs, honestly

Roughly ten to fifteen euros a year
For an ordinary name at an ordinary registrar. A first year at one euro is an offer, and the renewal is the real price.

Free on a Clay subdomain
name.clay.site is live, indexable once claimed, and never expires. It is not a trial address.

Your own domain on the paid plan
Pointing a name you already own at a Clay site is a change at your registrar and takes a few minutes.
What it looks like when it is live
Drawn rather than described, the page, the address, and what it can switch on.
Questions
Do I own the name forever once I buy it?
No. You hold it while you keep renewing, and it returns to the market if you stop. Setting the renewal to automatic is the whole of the protection.
Does a keyword in the domain help me rank?
Not meaningfully, and it has not for years. What it does do is make your address longer and harder to say on the phone.
Can I move my domain to somebody else later?
Yes, if it is registered in your name. That is the reason the ownership question matters more than the price.
Should I buy the misspellings too?
Only where the misspelling is genuinely likely and you can point it at the same site. Buying six variants of a name nobody mistypes is a subscription to nothing.
Can I start without one?
Yes, and most people should: a free website on your own address is live before you have decided on a name.
How do I choose one?
Say it out loud to somebody who has never seen it written: how to choose a domain name is the longer answer.
What happens to my old address if I add a domain?
It keeps working. Anything printed with it stays correct, which is why adding a name later costs you nothing.
Try it with your business
Search your business and see your site in seconds, free, no signup.