These days a true modern webdesigner should understand and be able to complete an entire website from scratch. While they may not necessarily be good at all parts, they need to know and be familiar with all parts so that they can be effective in a team or as a solo effort. This way they can develop things that are feasible, realistic, functional, useful, and good looking :)
What does this mean?
They need to understand business requirements, planning, staging, drafting, designing, implementation, maintenance, updates, upgrades and more.
Of all those things, they will need to know the business logistics, use case, functionality requirements, business requirements, in order to translate it into a usable design which accomplishes the goals mentioned previously, which of course would require some skill in design, from colors to layouts to photoshop or fireworks, ui, ux, and understanding of human nature and best practices.
From there on out, its taking the design, revising it, and putting it into html.
From there its implementing, developing and supporting the backend, so they should understand how a server hosts pages, how to roll out a server and services, how to serve the pages. Unless the design only calls for a completely static page, often they will need a backend to support a dynamic website such as php or ruby.
A good web designer will also have technical writing experience to provide documentation to customers and co-workers alike that explain how to use, how to update, how to maintain the work they've created.
This is not to say that they must excel at all of those things, but they should understand the concepts and workings behind them.
You should excel at one or two of these things to be a good web designer, if you excel at all of them you are an excellent well-rounded web designer and developer.
However even if you are the best designer in the world, and you know nothing about the technologies that power the design, you might as well be "just" a graphic designer. But to be a effective web designer, you design things that not only look good, but you design them with the understanding that it is more complex than just a graphic.