A wallet is software that stores private keys and allows interaction with blockchains. Your private key is the password to your crypto assets. It's a long random string that only you should know. The wallet stores it and uses it to sign transactions. Signing proves you own the assets and authorizes transactions. There are two types of wallets. Custodial wallets are run by exchanges or companies.
They hold your private keys for you. This is convenient but you're trusting the custodian not to lose your keys or steal your funds. Non-custodial wallets, also called self-custody, give you full control. You hold your private keys. No one else can access your assets. But if you lose your key, your assets are gone forever.
Popular non-custodial wallets include MetaMask (browser extension), Coinbase Wallet, Phantom (Solana), and hardware wallets like Ledger. Hardware wallets are the most secure because your private key never leaves the device. Your computer might be hacked. Your hardware wallet won't be. Custody is one of the biggest risks in crypto.
Many people have lost life-changing amounts of money by using untrusted custodians or by mismanaging their own keys. The tradeoff is between security and convenience. Full self-custody is most secure but requires responsibility.