Photos with more detail than reality.

...at least in terms of what the eye can see.

High Dynamic Range Imaging is a digital photography technique used to create photos with more dynamic range of luminance - basically the range of lights and darks - than would be possible with normal photography or even perceivable by the human eye in real life. In layman's terms, it creates a surreal, dreamlike effect. Most HDRI has very little or no areas that are washed out or too dark to see, meaning very little global contrast, so HDR can sometimes remove balance and negative space from the composition. HDRI looks amazing and is definitely a unique style, but shouldn't be used on every photo you take, it has its uses in certain contexts just like any other effect.

This is a pretty simple thing to do, but you aren't going to get HDR photos by going to the Filters menu in Photoshop and looking for an HDRI button. To make HDR images you have to be planning on it from the start. First, take multiple pictures of the same scene at different exposures, usually at +2/-2 intervals. To get a decent result you should have at least 3 pictures. Then upload the pictures to your computer and import them to software made to merge pictures in to HDRI - Photoshop works, but I find Photomatix gives much better results. Then once the images are merged you can adjust the tone mapping in different ways depending on the software you use. The best way to find out what things do is to just change them and see what happens, play around with the settings until it looks the way you want, then export it.