GIA
GIA is the Gemological Institute of America. This is recognised as one of the leading grading labs in the world and is a non-profit organisation. The world headquarters are located in California, USA.
Their website has some very good interactive diamond education tutorials.
AGS
AGS is the American Gem Society. It is half investor-owned and half jeweller-association owned. They have the reputation of grading the very best diamonds and hence you will pay more for an AGS certified diamond.
HRD
HRD is the Antwerp World Diamond Centre located in Antwerp, Belgium. It is a non-profit organisation. One of the departments is HRD Certificates. As a European-based lab, HRD certificates do not get much mention on the more prevalent US-based diamond informational sites. However, they do have a very good reputation in Europe itself, and are essentially the European equivalent of GIA. This laboratory was the first worldwide to gain accreditation itself for the quality standards of EN 45001, ISO/IEC Guide 25 and ISO 9002.
8 out of 10 rough diamonds pass through Antwerp, and 1 out of every 2 cut diamonds.
EGL
EGL is the European Gemological Laboratory. This is a network of institutes located in South Africa, India, Belgium, the UK, Israel, Turkey, Korea, and France.
IGI
IGI is the International Gemological Institute. This is one of the smaller laboratories.