Many different types of begonias are tender perennials and will overwinter in mild climates. From your description, I'd guess that the begonia we're talking about is one of the Begonia sempervirens-cultorum hybrids, also called wax begonias. They are the ones commonly sold in the cell packs with the rest of the bedding plants. I couldn't give you the commercial hybrid name, since there are several series on the market. For example, the Pizzazz series has a nice white (and the other colors) selection with those crisp green leaves.

I like the green leaved varieties the best, myself, but the bronze ones can be planted where there's a little more sunlight.

I hope it doesn't add to the confusion, but Begonia semperflorens-cultorum is often sold AS an annual and referred to as an annual. But now you know it as a tender perennial.