A nameserver is the server that holds the official DNS records for a domain. The domain's registrar records which nameservers to use, and the rest of the internet asks those servers where the domain's website, email and other services are. Whoever runs your nameservers controls where all of them point.
What it looks like
Nameservers are set at the registrar, usually as two or more names such as ns1.example-host.com and ns2.example-host.com. The names normally reveal the provider. By default they are the registrar's own, but they can be pointed at a web host or a specialist DNS provider such as Cloudflare.
Why it matters to a business
Changing nameservers moves all of a domain's DNS to a different provider at once, not just the website. If any record, especially an email record, is missing at the new provider, that service stops working. What happens when you change your domain's nameservers? explains the safe order.
When only the website is moving, it is usually simpler and safer to leave the nameservers alone and change the website's records instead.