There are times when you can assume that you will be the center of conversation -- for instance, in your studio during an open-studio event, in your booth at an art fair, at a gallery opening of your own work. However, those occasions are few and far between. It's a big world out there, and most people will have no idea who you are or what you do. Artists, like everyone else meet people all the time, and it makes sense to reach out to people you don't know or don't know well, finding opportunities to mix with people who might become buyers. For instance, one might join a town committee or the Rotary club or the Jaycees, volunteer at a church or public television station.
There is usually a "Hello, what do you do?" element of the conversation. Think, everyone is a potential collector, so it is wise to present yourself as a professional, with some prepared way of summing yourself up. For instance, "I'm a sculptor and do figurative works in bronze" or "I create large-scale outdoor metal sculptures on commission and for gallery exhibitions." It doesn't hurt to carry a business card or thin brochure with you that can be handed out to people you meet. (You don't want to hunt around for a pen and a piece of paper to write your name and phone number on, and you can't count on people you just met to type in your name and contact information on their cell phones.) Whatever you pass out should have your name, website and contact information; a brochure would have those things and include one or more images.
You don't want to say something along the lines of "I'm just an artist," because that sounds as though you want to get off the subject. If you mumble or tell a listener, "Well, I guess you could say I'm an artist," it sounds as though you don't know what you are about. There are some who will just say "I'm an artist," which is almost a conversation stopper, because it doesn't really tell people anything. Are you a painter, a sculptor? What kind of work do you do? Where do you show? Whomever you are talking to will get the sense that they need to play 20 Questions and pull the information out of you, which strikes many people as a lot of work, let's not go there.
If you say, "I'm a landscape painter, and I teach at Indiana University," the likely response will be a question about how long you have been teaching there or if you like the school, not about you as an artist and what you create. Direct people first to talk to you about you, and later on they may ask how you actually make a living.
There are two schools of thought about what you say to people. On the one hand is the short sum-up, which is often called the "elevator pitch" -- a capsulate version that won't take longer than the time it takes to get from the first to the fourth floor in an elevator. For instance, "I paint cityscapes, particularly images of my two favorite cities -- London and Paris. I'm having a show next month at the this-or-that Gallery, and there is an opening on the first night. I'd love if you could come."
The other approach can be described as more open-ended, tending to elicit follow-up questions and keep the conversation going. Ann Rea, a San Francisco landscape painter, for instance, tells people "I create an experience of art for my collectors." Colleen Attara, who calls herself an eco-artist, says, "I make joyful art from salvaged materials." (She might have described her work as "assemblage" or, more in the line of Richard Stankiewicz, "junk art.") Jenny McGee, a portrait artist, tells people that I "help people express their story through art." This approach identifies not so much what the artists do as it makes a case for the value of what they do and what is unique about them.
Both ways of identifying yourself and your art have worked for certain artists, and the effectiveness of each depends on one's comfort level with one or the other approach, but most important is having something ready at hand to say.