When graphic designer is faced with a new website design or logo design project, they have to think of the elements involved in the design. How important is typography, well, that answer depends.
Typography is the art type design, to include modifying and arranging the type, and the graphic designer has to adapt techniques in order to fit the required medium. For example, a paragraph introduction in a book or advertisement would not necessarily work on your website. Advertisements are held in your hand where as you are viewing a website on your computer screen (or smart device). This affects the layout for each medium dramatically.
Often, a designer will learn that when presented the job to change a logo, the typeface is the first element to work on. There are many sites such as www.dafont.com where a logo designer can enter the name of the company into the site which displays the business name in that desired font, then you may purchase the font. The use of white space (the space between elements such as bodies of text and columns) is important in creating a balanced design. However, too much white space will create an unbalanced design!
Typography can and will make your design successful, use it improperly and you may be subject to failure. Typography in web design is art and skill that takes time to master.