There are a few programs out there, some free, some pay-ware. Qtpfsgui is a free program that was popular, but I could never get the results to look right (colors looked surreal and gave HDRI a bad name). It is worth a look if you are interested in different techniques of tone mapping.
The HDRI process I'm now using is align_image_stack.exe and enfuse.exe which are part of the Hugin panorama package. These are command line programs which run on common operating systems. Thus, don't expect a cutesy graphics interface. In spite of them being command line programs, the process is very simple. I find the process simpler than Qtpfsgui which is uses a cute graphics interface. There are precompiled binaries for Windows and Mac. Enfuse is fast and produces pretty nice results. Both programs can handle 16-bit per color source images.
To produce HDR images, you need a digital camera with manual exposure mode, stable tripod, the two above programs, and an image editing program like Photoshop or The Gimp. A cable release or remote for the camera is useful to prevent camera movement when you snap the shot (using the self-timer setting also works). It's helpful if your camera can save pictures in RAW format so your contrast curves aren't altered by the camera. A dSLR is preferable, but a decent point & shoot camera will also work. Take a series of pictures at different exposures. The only thing you are allowed to change is the shutter speed. Each exposure shouldn't be more than 2-stops different. I usually shoot at 1.33-stop intervals. You may be inclined to change f-stops, but this will change the optical parameters which is a no-no. Your set of images should properly expose the dark areas through the brightest areas. Each image must be the same scene, i.e. a tripod is required and your subject must be static. After you take your images, you need to align each image so they overlay. Use align_image_stack to align the image stack if necessary. Even though you used a tripod, you may have slight movement of the camera between the various images unless you really have a solid camera mount. Align_image_stack will remove slight alignment skews. Next, run the images through Enfuse. The default settings for Enfuse work great most of the time. Run your final image through a photo editor to adjust contast levels, color balance, and cropping to your liking. It's so simple anybody can make their first HRDI in 10 minutes.
http://picasaweb.google.com/qrkpub/HDRIExperiments# are a few of my quicky experiments playing with Enfuse and align_image_stack.
http://qtpfsgui.sourceforge.net/ <--interesting but not my cup of tea
http://enblend.sourceforge.net/ <--enfuse tone maps really good
http://hugin.sourceforge.net/ <--comes with enfuse and align_image
While you're at it, play around with Hugin to make panoramas. The latest version of Hugin is simple to use and does a great job. No need to spend $300 for a panorama stitching program.