Relays are basically electrically operated switches that use low-power signals (from the Omega) to control higher voltage circuits, like lamps or home appliances. For the most up-to-date information, check out the article on the Onion Docs.
The Relay Expansion has two relay modules, you would be able to control two appliances in your home over the internet using the Omega.
The Relay Expansions are also stackable, so you can have as many as you want.
For more information and example code, see the following tutorial: https://wiki.onion.io/Tutorials/Expansions/Using-the-Relay-Expansion