The software I will be demonstrating for making HDR images is Photomatix Pro. There are other pieces of software out there, but I have found Photomatix to the easiest to use.
Take bracketed photos of your subject. Although not strictly necessary a tripod is recommended for best results.
Load up Photomatix and click the 'Load Bracketed Photos' button (hold down Ctrl to select multiple files). Only two bracketed photos are required, but three is better. Click OK.
On the 'preprocessing Options' dialogue window make sure the 'align source images' and the 'crop aligned images' boxes are checked. I generally find the align by 'Matching features' to be the best option, but if you find the final image is not aligned try by 'horizontal and vertical shift'. (If you have used a tripod there should be no alignment issues).
If you have been using a high ISO setting you may wish to check the 'reduce noise' box, and if you have any ghosting or chromatic aberrations check the relevant box. Click OK and the software will do its thing (this can take a few minutes on older computers).
This stage of the process is difficult to teach. I would recommend choosing a result from the preset pictures at the bottom that looks close to your desired and then tweak it from there. I usually find that the colours are a little over saturated so I decrease the saturation and luminosity by a small amount (make sure you have the 'preview' box checked). The best way to learn about what the individual settings do is just to play around with them and get a feeling for how they affect the picture.
When you are happy click the 'process' button. (Again this stage can take a few minutes on slower computers).
The final result is a blend of the 3 original pictures with the foreground, subject and background all correctly exposed. It is worth noting that if anything moves within the frame while you are taking the bracketed photo there will be some blur in that spot in the end result. This has been demonstrated by the leaves in the top right corner which were blowing around in the breeze while the bracketed photos were taken.