HDR (High Dynamic Range) rendering simulates brightness above that which a computer monitor is actually capable of displaying. This mainly involves "blooming" colours above 100% brightness into neighbouring areas, and adjusting a virtual camera aperture to compensate for any over-exposure that results.
Beside the obvious effects of this (see right), HDR gives richer colours and finer gradients: since bright and dark areas are pushed into white and black, correctly-exposed areas are drawn with a far wider range of values.
Source does not have a physically accurate HDR simulation. Its camera has a far wider range than the human eye (let alone real cameras), and it also adjusts to changes in brightness far faster. Both of these are expedient to gameplay of course, especially in multiplayer.
In fact Source does not even create 'proper' HDR images: instead it collapses the image down to LDR early. The average user would be hard pressed to tell the difference however, and the benefits include support for all DX9 GPUs, MSAA compatibility, and excellent performance.