"Web design" is one of those industries that, over time, has developed into separate disciplines without the public at large being aware of any other term to use when referring to it. As a result, there are nowadays web designers who only do web design (and tend to look at you funny if you expect anything else), as well as "web designers" who do a bit of everything you can possibly think of that's related to websites, and of course some who do a bit of something in between.
As a prospective client, it's important to know exactly what you need your "web designer" to do. Once you know the difference between each role, you can hopefully read on their website (or ask in an email if they've somehow overlooked to mention it) exactly what services a freelancer or company are offering.
Web Designer vs Web Developer
The biggest distinction in the industry is between designers and developers. Simply put, designers come up with the look and feel of a site, create all the snazzy graphics, and then a developer comes along and actually builds it. There are those who perform both sides of the equation, and there are others who do not, so never assume that "web design" involves the full build unless it's specifically stated.
Each of these two roles is split into further individual positions, but unless you're working with a big budget and need to specifically hire people for each area you're unlikely to worry about these too much. Some examples are Front-end vs Back-end Developer, plus odd sounding specialities like User Experience (UX) Designer.
You may also run across developers specializing in one particular platform. A good example of this are WordPress developers, since WordPress sites alone have become a huge industry in recent years. If this is what you're looking for, you may be disappointed (but by this point not surprised) to learn that even these come in different types. WordPress theme developers and WordPress plugin developers are often completely different people, so that's just one more thing to ask when weighing up your options.
Logo/Branding Designers are not Web Designers
And vice versa. A logo is an integral part of any new website, and the site should always be designed to fit the existing logo. Branding as a whole also goes way beyond just a logo, involving print design for letterheads and business cards among much else.
The two forms of design have very different goals and technical requirements, and therefore a very different skillset, so while you may find some people who do both of these things well you shouldn't just assume you'll find them in the same place.
So what does a Web Designer actually do?
At its simplest, web design (like all design) is about communication. The goal of most website designs is to present information to the user in a clear and intuitive way.
You may have heard the phrase "content is king", and this is especially true in web design. A good web designer should always get at least a good idea of the sort of content that will go into a site before starting to design, since a pretty site without enough room for the text will end up a mess.
Stage one of website planning is the wireframing stage. The wireframing technique varies from designer to designer but the goal is always the same: to work out the most efficient possible presentation layout for the information. This stage is quite dull from an outsider perspective and clients can sometimes become frustrated, but think about how important it is that users can actually find what you're trying to sell them (or whatever you're trying to get them to look at) and hopefully wireframing will seem just that little bit more worthwhile.
The "fun part" is of course making the graphics for a site design and pulling it all together into something resembling the finished product. Popular tools of the trade are Fireworks, Photoshop and Illustrator, all Adobe products, though Fireworks is actually geared toward web design specifically and is probably the easiest to learn from scratch. There are also free software options, such as the GIMP, which can with practice serve just as well.
A large part of designing a website is of course knowing what can actually be built. If a client approves a design only to be told by their developer that it's impossible in reality, they won't be very happy at all! For this reason, many web designers have at least a bit of front-end development knowledge which involves the very basic languages like HTML and CSS. As with most things we've discussed, whether they actually offer these as services is something that varies from person to person or company to company.
Do I need a Web Designer, or a Web Developer, or both?
Most small sites, such as for small businesses or individuals, are designed and built by the same person or company. The distinction between roles really plays a major part on larger projects, or in full-blown agency settings.
The advantages of hiring just one person to deal with your entire site are mostly communication-based. Especially if you're hiring a freelancer, you'll probably end up with a friendly working relationship rather than dealing with multiple people for different aspects of the project. They also run the entire project themselves, and are not waiting on others to complete their tasks before being able to continue.
On the other hand, if you're looking to have only one aspect of your site reworked (for example a new feature coded in, or a particular page layout redesigned) it may be worth considering someone with specialist knowledge in that exact field. If your project is more than one person can handle, having tasks split into each discipline can actually be very advantageous and avoid multiple people waiting to work on the same thing.
In the end, it's down to your judgement which type of professional you need for your project, but hopefully you now have a better idea of the skills you're looking for and what to ask.