An output descriptor (note that output descriptor and wallet descriptor refer to the same thing) is a human readable string that represents an output script (a scriptPubKey) and everything needed in order to solve¹ for that script. Descriptors also have a bech32-like checksum which allows for the descriptor to be given to others with less risk of accidentally mistyping or losing some characters in the descriptor string.
Descriptors are unambiguous as to the public keys to use (derivation paths for extended keys are explicit) and the scripts to use. This makes them suitable for importing to other wallets without confusion. In contrast, traditional import mechanisms support only keys with special versioning to indicate the scripts to produce, and don’t provide the derivation paths. This creates a situation where a user imports an extended key into a wallet but is unable to see their addresses because that wallet uses a different derivation path than the original wallet. Descriptors avoids this issue entirely by specifying the derivation paths (if any) and the scripts to produce.
A descriptor wallet is one which stores output descriptors and uses them to create addresses and sign transactions. By abstracting away address creation and transaction signing to a largely standalone module, such wallets can upgrade to using new address types much more easily.
For most users, the only visible effects will be in wallet import/export. Descriptors will only be shown during exporting, and descriptors should only be handled by the user when they want to import their wallet. Wallets that use descriptors internally shouldn’t have any noticeable effect to the user.
¹: Solve – Solving a script means that one would be able to create a final scriptSig/witness with valid signatures if they had a private key. This means that all public keys and other scripts are available.