"Atomic Swaps" allow two parties to exchange tokens from two separate blockchains without any counterparty risk and without the need to trust each other or a third-party (like an exchange). In its most basic form both parties create transactions to their trading partner in a way that either outputs from both transactions or none of them can be spent (thus making the exchange of both cryptocurrencies atomic).
Since the activation of SegWit on the Bitcoin network and the upcoming availability of the Lightning Network these types of swaps no longer have to occur on-chain obligatorily, but also can be carried out via the second layer Lightning Network if both chains support it (in fact the design of the Lightning network explicitly considers and enables these types of cross-chain exchanges).
This talk shortly outlines the history of the idea of atomic cross-chain trades, how to construct them and what features a blockchain needs to support these and how they will work in the context of the Lightning Network.