What is a domain name, and how is it different from a website?

The short answer
A domain is your address on the internet — the part someone types to reach you. A website is what they find when they arrive. They are bought separately and can be lost separately, which is why owning your domain matters more than almost anything else: a website can be rebuilt, but a domain someone else controls cannot simply be recreated.
On this page
The distinction
A domain is an address. A website is a place. Someone types the address and arrives at the place.
That sounds pedantic and it is not, because the two are owned separately, renewed separately, and lost separately. Most website disasters are really domain disasters.
Why the domain is the important half
Everything else can be rebuilt. Lose your website files and you can build again — expensive and irritating, but possible. Lose control of your domain and every link anyone ever made to you, every printed card, every search result and every saved bookmark now points somewhere you do not control.
Which is why the most useful thing in this article is the next section.
The one thing to check today
Is your domain registered in your business's name?
Look it up on any WHOIS service, or ask whoever set it up to send you the registrar login. If the registrant is your developer, an agency, or a former employee, get it transferred. It is a routine request taking a few days, and it is far easier while everyone is on good terms.
What it costs
A standard domain is inexpensive — an annual fee in the low tens. Country domains and premium names cost more.
Two traps. Introductory pricing: a first year at a token price can renew at several times that, so check the renewal rate rather than the headline. Expiry: a lapsed domain can be bought by anyone, and recovering it afterwards ranges from expensive to impossible. Turn auto-renew on today.
Choosing one
Short, sayable, and spellable by someone who has only heard it aloud. That last test rules out most clever spellings.
Avoid hyphens and numbers where you can. They survive on a business card and fall apart over the phone, and "is that with a dash?" costs you visitors every time.
The ending matters less than people think. A .com is the safe default; a country domain is a good signal if you serve one country. Neither decides whether you rank.
Who should hold it
The check in this article is the important part, so here is the standard to hold any supplier to: the domain should be registered to your business, with your email as the registrant contact.
A supplier managing the domain is normal and fine. A supplier owning it is not — and the difference only becomes visible on the day you want to leave, which is exactly the wrong moment to find out. Helm registers it in the client's business name as standard, which is less a feature than the only version of this arrangement that survives a parting of ways.
If yours is currently in somebody else's name, it is fixable, and it is worth fixing now rather than during an argument — see someone else controls my domain.