Atomic cross-chain trading (Atomic swaps is another name) is, for example, when you hold, say, BTC and I hold LTC and we decide to trade directly without a centralized exchange to serve as a mediator.
The problem is, of course, who sends the coins first. Can the sender trust the other person that they will send their coins back without cheating, or, for example, without chaning the agreed upon exchange rate.
It's fine when you trade with someone you know (like it happened recently when the creator of Litecoin successfully traded LTC for BTC, VTC, and DCR), but what about when you don't know the other person? Hence a trustless mechanism is needed.
In principle, these swaps rely on two mechanisms: either on using time locks or on using contracts.
More info:
https://en.bitcoin.it/wiki/Atomic_cross-chain_trading