It's pretty much the same division as "design" vs. "engineering" in any sort of product development. Design concerns what a product does and how it looks and feels to the user of the product.
Engineering makes the product do what it's intended to do, without unpleasant side effects, and that it can be manufactured at a cost that makes sense.
Very loosely speaking, design makes a product worth buying and engineering makes it worth selling.
For a website, the designer creates the part of the website that the user sees: what's on a page, how it's laid out, how the pages work together--creating pretty much all of the CSS and the majority of HTML on a page, finding/creating images, sounds, animations and such, as needed.
That's a thumbnail sketch, of course, and there's plenty of room for someone to quibble. There's plenty of crossover in practice, particularly in a small team. What's important, if you're looking for a possible career, is to look at what *you* enjoy doing.
I had the opportunity several years ago to help out with a rapid prototyping program offered at two nearby high schools. One of the speakers, a designer from Ford, stressed this to the kids. If you like making things look cool (or hot, or whatever) get into design. If you like nuts and bolts and making things work, get into engineering. A left-brained engineer-type won't be happy working with touchy-feely things; and a right-brained designer won't like the "irrelevant" details needed to make a product both functional and buildable. If you plan to spend a quarter or more of your entire waking life doing something, pick something that won't drive you crazy.