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.
Manages keys, not coins
Can be custodial or self-custodial
Hardware isolates signing
Backups are essential