Cryptocurrency wallets are generally categorized by the entity which has control and stores the private key for the wallet. Below are common examples:

A private wallet is a wallet were the end user is the only one who has access to their private key. This type of wallet puts all the risk and control of assets in the hands of the user and the wallet client they are using.

A custodial wallet is one where an end user has no control of their private key. The owner of the service they are using holds a private key controlling all user deposits and withdrawals. The user’s balance is typically tracked on a separate ledger to the on-chain one and the real cryptocurrency value is stored on a wallet owned by the company. This type of wallet moves the risk from the end user to the company controlling the wallet.

A hybrid wallet tries to incorporate both a private and custodial wallet. The user can choose if they want to take on the risk or trust a custodial service to manage some of their funds.

While not in the same category as the above it's worth mentioning hardware wallets, which are essentially private wallets. Hardware modules or wallets are secure devices built for holding private keys. The device is used to sign transactions using the private key without ever exposing the private key. These are often accompanied by a paper wallet backup.