Web design is a web service, used in web projects that require custom and unique design. Not all projects require an unique web design – some could save a lot of money and use a free website template, while some other could work well with a paid template (but not unique). Every option still has a need for some unique graphics, images and photo processing, so you still need web or graphic design.
Unique web design is used by all corporative web sites (medium to large sized). They all want an unique branding, so they are widely recognizable by their audience and potential customers.
Some startups may also need unique web design, so they can meet their goals. If your project is big, the web design part is usually a (small) percent of it.