A resolver is a contract on the Ethereum network that answers questions about the name, such as which address it points to. It stores the name's records: addresses on Ethereum and other chains, text records such as avatar and website, and content hash.
Permissioned and custom resolvers
Most ENSv2 names use the Permissioned Resolver, the standard ENS resolver. To see which resolver a name uses, open the name in ENS Explorer and select the Resolver tab.
A custom resolver is any other contract, and is for advanced users. ENS Explorer doesn't check that a custom resolver works, so a wrong address means records won't resolve and no error shows.
For how a resolver differs from a registry, see Registry and resolver: what's the difference?
Common questions
Why do all my names share one resolver?
The first time your wallet gets an ENSv2 name, by registering or upgrading, a resolver is set up for it. Names you get after that use the same one, unless you change it. Each name keeps its own records, but who can edit them is shared: a wallet you give resolver roles can edit records on every name on that resolver, and on any subname with no resolver of its own.
To give a wallet those roles, see Grant resolver 'roles' to edit name records.
How do I give one name its own resolver?
Open the name's Resolver tab in ENS Explorer, select Change resolver, turn on Deploy new permissioned resolver, then select Save changes and approve both requests in your wallet. The steps, and what to check before you use a custom resolver, are in Change the resolver on your name in ENS Explorer. The name's records don't come across, so set them again.