Web fonts are standard and come installed on all computers (you can search for a list). They are fonts like verdana, georgia, etc, and are usually called web safe fonts.
I have never heard to other fonts called "graphic fonts", but these fonts are not previously installed on computers and must be downloaded. If the font is used in web design, you'll have to use CSS to display it.
Web fonts are fonts embedded using @font-face.
Haven't heard of "graphic" fonts, but I'm guessing you mean like icon fonts. Windows has a few installed already, like Wingdings, but you can make your own with any of the several online tools out there.
Web font consists of tuned font and use for developing the professional websites & website design by using the cascading style sheet.
Web font and Desktop font is not same font because desktop font is used or for software applications of computers.