In general, it is writing that promotes a person, opportunity, idea or opinion. Since we're discussing websites, we will just assume the writing will be on a website (and this will be referred to as search engine optimization copy).
There are definitely other mediums that make use of copywriting and copywriters; television ads, radio commercials, billboards, bus ads, newspaper ads, magazine ads, etc. You come up with the idea and the point behind writing copy.
It is particularly designed to persuade, entice, motivate a listener or in the case of a website a reader to take action. That step could be anything from buying a product to subscribing to a newsletter, from sampling a product to participating in a survey or poll.
By the way, you can also make use of copywriting to persuade a reader to not do something or not trust something.
What types of materials make use of copywriting? Body copy, headlines, slogan, direct mail pieces are biggies, taglines, jingle lyrics, slug lines, scripts, captions, the World Wide Web, news releases, research and white papers and Internet (website) content. It's a pretty extensive list, isn't it? And it's just the tip of the iceberg too.
Don't forget there are also things like print ads, mail order catalogs, brochures, postcards, greeting cards etc. Copywriting is an excellent line of work to be in if you are any good at it, as there will always be work.
When used on websites, copywriting usually refers to the method or style of writing and wording content that is slanted to achieve higher rankings with the search engines. This is actually known as content writing which means the proper positioning and repetition of keywords and keyword phrases on a website.
At some time, writing content for websites served as a means of getting better rankings because the writing was geared to the search engine algorithms. That is no longer the case and writing is now more geared to human visitors and search engines.
Another form of viral copywriting can be in putting a lot of content on other people’s blogs. In fact, you can pay to have your product promoted on different blogs for a set fee. You may not have to choose the copy related to your link, but it can be a straight mention on a blog that may have hundreds, or even thousands, of readers. The trick with this type of writing is to produce fluent and readable copy/content written with an eye to search engine optimization.