Sorry, bobince and Diogenes, but I must disagree with you in the strongest way possible in regard to not using pt sizes and always using pixels:
A designer wanting a font size to be relative to the default size specified in browser or OS settings, which should be the standard, should use % or em. A designer wanting an absolute size (on screen) should use pt to keep the font size constant regardless of the screen's DPI setting, and here's why:
Modern laptops and mobile devices' screens have much denser pixels than desktops', up to approximately 150dpi of natural (default) resolution compared to the desktop's 72. Medical and other specialized screens go even higher. And the user has the ability (if not always the knowledge) to raise or lower the logical dpi to customize the display according to his preferences and sight ability.
Any image or text sized in pixels will be too small to read on a dense-dpi device, making your good-sighted users struggle and rendering any users with less than perfect sight unable to read the text. More importantly (and one can argue if this is correct behavior or not), some browsers do not resize pixel-sized text even when the user commands it, making the situation unfixable.
Unlike paper medium, where position to the precision of 100ths of an inch is possible and everyone reading a designer's work sees the same-sized paper, the diversity of devices used to access the Net invalidates the entire concept of pixel-perfect design, which looks good only on the one device it was targeted at. The point of the design on the Web is to make the site accessible to the target audience, which usually should be as large as possible and would be using a range of devices and settings.
Operating systems are already going there with vector UI elements or, at the least, large source bitmaps that are scaled smoothly. So should you, forgetting about the arbitrary unit of measure called "the pixel" and making your designs compatible with the widest array of access devices possible.
Addendum - to additional points raised by Diogenes
Unfortunately, the real-world situation with DPI awareness (and thus with proper support for pt-sized text) in currently popular operating systems is, for the lack of a stronger printable word, a clusterfail, as summarized by the CSS-Discuss Wiki:
Unfortunately, most computers don't know [the display's DPI], so the browsers don't either... In the absence of this knowledge, operating systems and browsers typically just make assumptions, and these assumptions are usually wrong.
So, while in theory points would be the best unit for fixed text, in practice using them today produces inconsistent results in many OS/browser/screen configurations. Thus designers often have to give up and revert to pixels to achieve fixed font sizes. Personally, I use percentages and ems, avoiding fixed sizes altogether.
The CSS-Discuss archives have a great summary of the font sizing issues, and numerous links to details and various approaches.