So I don't think of myself a great networker. However, my SO is an absurdly good networker. It's hard to explain, but it's always astonishing what a great job he does of making use of the connections that he has.
When I've asked him about it, he's explained that it's really important not to think of any given interaction with a person as "networking." He does take an interest in getting to know people who are involved in his field, or might be helpful connections in the future, but he does so in a very casual way - the same way you'd make any other kind of friend. You meet someone, you like them, you get their card, and suggest that maybe you get a drink sometime. Don't immediately start treating them as a resource.
Another thing he does that I am totally impressed by (and am, again, terrible at doing myself) is that he tries to think of connections that he can make between people he knows that will have no particular benefit to him at the time. He meets an entrepreneur, he says "hey, let me introduce you to a VC I know." He meets a blog writer, he says "Hey, let me introduce you to this publisher I know." People are really impressed when one of those connections actually pay off, and they start to think of him as someone who knows people, and gets things done. Obviously, this is a tall order, and involves a kind of constant long-term strategy, but it's kind of amazing to observe.