Cold storage vs. hot wallet
This distinction is about internet connectivity specifically — a related but slightly different question from the custodial vs. non-custodial split covered elsewhere on this site.
Hot wallets: connected, convenient
A hot wallet — a mobile app, browser extension, or exchange account — stays connected to the internet, making it fast and easy for everyday transactions. That same connectivity is also what exposes it to remote hacking attempts, phishing, and malware in a way offline storage isn't.
Cold storage: offline, harder to reach — for attackers too
Cold storage keeps private keys entirely offline, commonly on a dedicated hardware wallet or even a piece of paper. Because there's no network connection to exploit remotely, cold storage is generally considered far more resistant to hacking, at the cost of convenience for frequent transactions.
A common split: most in cold, some in hot
Many experienced holders keep the bulk of their holdings in cold storage for long-term security, and only a smaller, "spending" amount in a hot wallet for active trading or transactions — similar in spirit to not carrying your entire net worth as cash in your physical wallet.
Cold storage isn't risk-free either
A lost or damaged hardware wallet, or a lost seed phrase backup, can mean permanent loss of funds with no customer support to call — cold storage trades hacking risk for a different kind of risk: your own record-keeping and physical security.