Ecommerce (or e-commerce) is the art and science of selling products and/or services over the Internet.
While some prefer the narrower ecommerce definition of conducting financial transactions through electronic means, I prefer the broader ecommerce definition because there are still a lot of people trying to sell things over the 'Net who aren't actually conducting financial transactions electronically.
There have been many discussions and outright arguments about just what e-commerce is (and isn't). If you like this sort of thing, "The Evolving Definition of E-Business" is just one example of the many papers written that attempt to elucidate a more rigorous ecommerce definition.
E-commerce may be conducted B2B (business to business) or B2C (business to consumer). But simply put, any website that attempts to make a profit is involved in ecommerce.
You could easily write an entire encyclopedia about all the related terms that ecommerce has spawned. Merchants who conduct business on the Internet are called 'etailers', for instance, and sell merchandise through 'cyberstores'. And who will ever think of 'shopping carts' in the same way again?

