Chess fonts are used for diagrams or figurine notation. Some fonts also have symbols for comments (like +- etc.) and others have symbols that can be used for illustrations or chess clipart.
The advantage of chess fonts is primarily that it is easy to make diagrams if you have a pgn file - and there are many font designs available. Also the diagram takes up very little space, because it is text and not graphics.
On the page Chess Fonts FAQ is advice on how to use chess fonts and solve known problems - problems often made by so-called "intelligent" word processors.
A chess font is a font just like Arial, Helvetica, Times etc., but instead of ordinary letters like A, B, C (or besides) the letters are replaces by chess figurines or symbols. If you have selected a chess font in your word processor and press a key (e.g. K), you get a figurine on a square and not a letter. In that way you can build a diagram or write games with figurine notation.
In principle a diagram is exactly like:
**********
*AAAAAAAA*
*BBBBBBBB*
*CCCCCCCC*
*DDDDDDDD*
*EEEEEEEE*
*FFFFFFFF*
*GGGGGGGG*
*HHHHHHHH*
**********
But instead of stars and letters you get squares with or without chess figurines. The chess figurines are squares (same height and width) so when pieced together you get a diagram. Very simple.
To get going you need to install a chess font, and on the page Chess Fonts are several free fonts. A new font is easily installed from the control panel, and the procedure is described in Window's help file.
Making diagrams manually with chess fonts requires time and patience, but fortunately there are programs, which make things easy. In commercial programs like ChessBase you can play through a game and pick the position of which you want a diagram. There are freeware programs that can do the same thing - and they offer a wider choice of fonts than e.g. ChessBase. You can find these programs on the page Chess Diagrams.
These programs typically save or copy the "text" - the diagram - as RTF (Rich Text Format) which almost any word processor can read. That allows you to paste or insert the textfile into your document, and you've got your diagram. Other programs embed the diagram as an object (OLE), and there are Word macros that can create a document with diagrams from a pgn file.