In summary:
With custodial or "normal" exchange trading, you send over your funds to wallets these exchanges control. Making trades instantaneous, but leaving your funds more susceptible to hacks.
On the other hand, with non-custodial or "defi" exchange trading, you're going to use and connect your non-custodial wallet(e.g. Metamask, MEW, etc) to these non-custodial exchanges for you to be able to exchange your coins/tokens. Downside being you're going to make an on-chain transaction after every trade(which can be slow and costly, because fees), but with an advantage of putting your funds at less risk of hackings.
The explanation above is clear and simple to understand. If you wish to read further about DeFi and the related exchange platforms and how they differ from the rest you could do so
here.
This is a well written article that I think rounds up several points new people need to be aware and understand about DeFi and how it fits to the rest of the crypto market.