It used to be the case that the internet was pretty dull when it came to typography. System fonts such as Arial, Times New Roman, Verdana and Comic Sans (for those with a bad sense of humour) being commonplace and the only functional font faces that worked cross browser, however web typography has come leaps and bounds in recent years, mainly because of the development of new browsers, new standards in CSS and standardised font file formats.
A number of foundries old and new have begun to provide web fonts in a variety of formats at a cost, but a few have also started to provide open source typefaces.
Google Web Fonts has a selection of hundreds of fonts stored on its servers which can be added to a website using its API, these fonts even function on earlier browsers such as IE6.
Font Squirrel is another service which offers a wide selection of free fonts for commercial use which can be embedded onto a website in a number of different formats.

With the introduction of CSS3 and cross browser standards, type can also be formatted in a number of different ways (kerning, leading, weight etc). This is beneficial for designers as it gives you a lot more options to play around with, it is also beneficial for search engine optimisation, because previously copy with a nice typeface would have needed to be converted to an image.
Typography is an important aspect in web and graphic design, and at By The Scruff we have also started to incorporate these new web fonts into our designs, for a recent example check out: https://myarchitect.net/
Check out these links for more interesting online typography: