in 1984 (a book by George Orwell, published in 1949) Big Brother, is the leader of the political party in power. The world is a dystopia with posters that literally say "BIG BROTHER IS WATCHING YOU"
The party controls everything- it controls when your TV turns on, so you can do morning calisthenics (and there's a camera so they can watch you do them). Everything is monitored, and moderated. There is an entire ministry dedicated to altering records as the party sees fit (like when Stalin removed Avel Enukidze from a picture).
A bit of speculation- Big brothers have certain roles in western society- one of which is protector, but not in a Oedpial way (like a father would be), which is probably why Orwell chose that instead of "FATHER IS WATCHING YOU." The implication is that "you are being spied on because I know what's best for you" and it's part of the propaganda strategy for the Party to make you feel ok with being spied on, but also to remember that you are being spied on, so you better behave (sort of like putting fake security cameras everywhere)
It's a good book, and a short read, you should check it out, so I won't spoil things about the plot.