An update, several years later:
What technology to use for integrating R into web apps depends upon the scale of what you are trying to accomplish.
For small, standalone web apps, Shiny is fast becoming the de facto standard. It is relatively easy to use, and there are paid support options.
For bigger projects, you are probably better off using whatever web framework you are comfortable with and then figuring out how to connect to R. (You can call R from the command line from any more-or-less any other software, and there are some slightly nicer interfaces to R such as rJava for Java apps.) This sounds a little vague, but for such projects the important decisions are based on "what does your company already use?" and "what do you need R for?"
Original answer:
There are lots of different ways of creating a web app with R. Most (if not all) are listed in the FAQ on R document.
If you want to use RApache, there's a presentation on the old RApache site that might be useful to get you started. Otherwise, RWui is very simple to use, but not so flexible (last time I used it, anyway), and rcom/StatConnector can be used to run R from your web server.
