Short answer
With a custodial wallet, a company holds the keys and you hold an IOU. With a self-custodial wallet, you hold the keys and the coins sit on-chain in your own address.
Exchanges are the classic custodial example: you deposit, they keep the coins in their own wallets and show you a balance. That's convenient, and they can usually reset your password, but you depend on them staying solvent, honest and willing to process withdrawals.
A self-custodial wallet removes that middleman. Trades settle straight from your address on the blockchain, and anyone can verify the balance. The cost is that recovery is your job, not theirs.
Neither model is risk-free. Custodial risk is about the company; self-custodial risk is about you losing keys or signing something malicious. For on-chain memecoin trading, self-custody is the norm because the tokens live on decentralized exchanges, not inside an exchange's order book.
See the checks for yourself
18+ only. Self-custodial: you keep your keys. 0.5% per trade, no minimum. Memecoins can go to zero.