My definition is that social networks are a series of sites where people can pretend like they actually know someone. You go, add people as "friends" even though you've probably never met them, may never even talk to them, and really know next to nothing about them. The sites encourage people to share all kinds of personal details so that they can then pass this along to marketers who have salivated over that kind of info since time immemorial, but have always had pesky privacy laws and regulations, forcing them to resort to gimmicks like free or low cost magazine subscriptions, some kind of store discount card, etc. Now they just offer up some trinket like a farm sim game and people will willingly hand over just about anything.
Social networking is a farce. It's not social, it's barely networking in the non-computing sense. It's a bit crude, and I'll clean it up a little bit, but I saw this somewhere else and thought it quite apt. Social networking is kind of like calling masturbation the same thing as sex.
But a less cynical view would be that social networking is essentially the same thing as the age old business networking. You meet someone, and that someone may know someone, and so on. So if you're ever looking for a job, you can call up the first someone, who may call the second someone on your behalf, etc. You could think of it like an extended family, only with people you're not related to.
Computer networking is a means of connecting a series of computers together such that they can communicate and share data. The Internet is really a pretty ordinary computer network, just on a massive scale. Usually you'd have only a few thousand computers at most on a network, the Internet is hundreds of millions at least.