If you want to control your home using an arduino,
You need a webserver...
Here you have 3 options...:
Buy a server (buy a dell, hp etc. server, write some php script and run it as a "website")
Use a server host (many dedicated or non-dedicated server hosts out there)
Use aws (amazon web services is a great way to get a free server going if your not goin to do much with it, free for one year!)
And you need some relays, i would not reccomend connecting the relays to your power outlets tho, use some king of home automation remote and power jacks and hack it.
Ps. If your gonna host a webserver make sure u got a firewall and some security so your lights dont go on and off every few seconds!