A Bitcoin wallet does not contain coins. Coins remain represented by UTXOs on the network. The wallet stores or coordinates signing keys, identifies relevant outputs and creates transactions.
A custodial wallet lets a provider control the keys and honor account withdrawals. A self-custodial wallet gives the user direct control and direct responsibility. Multisignature arrangements distribute authority across several keys.
A hardware wallet isolates signing keys from a general-purpose computer. Its screen should independently show the destination and amount. Hardware reduces some attack surfaces but does not replace careful backups, verification or supply-chain judgment.
For the clearest picture, read this entry together with Private Key, UTXO, Transaction, Full Node, Lightning Network. The reverse links also lead from UTXO, Private Key, Lightning Network, Bitcoin address.
Manages keys, not coins
Can be custodial or self-custodial
Hardware isolates signing
Backups are essential