1. I mostly just knew I was a furry after learning what a furry was and realizing I acted the same way/liked the same things, just without the term attached to it. That was about 3 years ago or so.
2. I don't normally tell people I'm a furry because, in general, people either 1) have the wrong idea about it, or 2) have no idea what a furry is. Unless "furry" comes up in conversation otherwise, it's hard to know how people think (or don't think) of it, so I don't like to be the one to bring it up. A couple people know I'm a furry, but most don't. Though if they came right out and asked me if I was, I'd most likely tell them yes.
3. I love drawing anthropomorphic characters. It has helped me a lot in improving with art and such.
4. The only way you can really define being "furry" is that you have an interest in animals and/or anthropomorphic characters. Beyond that, pretty much everyone has their own definition of what being furry means to them. There are several different aspects to the fandom, and not everyone is necessarily into the same parts of it as everyone else. The biggest problem we have is when people generalize the fandom and say "All furries are like [this] and they do [this, this, and this]. It's that simple." We've formed a community based on a common interest, and naturally there are different kinds of people within that community. We just ask not to be judged before you get to know us, just like any other person.
I would highly suggest reading Wikifur (which Timmy linked to above) for more information. Also, there is another thread here where our forum members have talked about how they became furry, which you can find here:
http://www.thefurryforum.com/forums/index.php?topic=5311.0