From Bulbapedia, the community-driven Pokémon encyclopedia.
Chain breeding is the process of breeding a move to a Pokémon that can only learn it as an egg move, then in turn using that Pokémon to breed it to another Pokémon that can learn the move as an egg move, but cannot breed with the original Pokémon.
An example of a move that is required to be chain bred would be Outrage.
It can also be used to pass on entire movesets, even if individual moves are able to be passed down without an intermediary. For example, taking the child Larvitar with Outrage and leveling it up so that it will learn Thrash, it can pass down both moves to a child Gible, which cannot learn either of them normally.
Many chains, however, are not as short as this Outrage chain, with a few having several steps that are required to be followed. What is required of all chains, however, is that the child is a male Pokémon so that he can continue to pass on the move. Several chains even require that a male Smeargle Sketch a move specific to a Pokémon of an all-female species, like Miltank, in order to pass it down.