NOTE: this answer was added in 2011. Much has changed since then, but most of these recommendations still hold true. —Mike
You can think of web design the same way you think of poster design (in most cases). You have a short period of time to engage a user, deliver the message and hook them in to spending more time to find out the details of your site. That's the biggest difference between designing for the web any anything else.
As far as composition, your canvas will need to adapt to screen sizes from 320px wide to 2100px wide, which may call for completely different designs based on how much content can fit. These are some rules of thumb, but it really depends on your audience.
You will also need to consider how your audience is viewing your site. A site rendered in Internet Explorer may not have the same qualities as the latest version of Safari or Chrome. You will have to decide how your design will gracefully degrade to comply with these older browsers.
As far as type faces and sizes, you shouldn't go below 9px in any instance, with body text not falling below 12px, though the user has the ability to change this on their end. Font families are the norm on most sites. Even if you use TypeKit (font embedding), you still need to define a set of typefaces that everyone has as a fallback, e.g. Times New Roman or Georgia. See: http://www.awayback.com/revised-font-stack/
Finally, you have to consider the interaction a user will have with the site. What will be their gut reaction when visiting the site? For example, make sure your primary navigation is able to be easily differentiated from other elements and looks and feels like a navigation. Also, the page identity is usually found in the top left, but as long as it's near the top you should be fine.
There are millions of other things to consider before even starting to think of the design, but I hope those are a few that can help get you started.