You'd need the relay expansion for that:
Relays are basically electrically operated switches that use low-power signals (from the Omega) to control higher voltage circuits, like lamps or home appliances.
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.
From here: https://onion.freshdesk.com/support/solutions/articles/6000067410-what-does-the-relay-expansion-do-
And also this: https://onion.freshdesk.com/support/solutions/articles/6000082229-how-can-i-use-the-relay-expansion-
In terms of controlling our Omegas remotely, well they're still working on getting us the apps and libraries to make that kind of thing easier. In the meantime, if all you've got is the expansion dock, you could start playing around with GPIO? I'm new to this and just having fun with that so far. Here's a good intro if you're new: http://www.instructables.com/id/Blinking-a-LED-with-Onion-Omega/