A Laser Inscription is usually a series of numbers that denotes the report number that goes with that stone.
In the case of the GIA laboratory, the inscription is always on the Girdle of the Diamond (the small section of the stone in-between the Pavillion facets and the Kite facets). The stone will always show the GIA logo followed by the report number.
The laser inscribed number on the girdle cannot be seen with the naked eye and can just about be read under 20x magnification or loupe. Even in this case, a practised eye and a steady hand will be necessary. For the benefit of the image below, I have taken a photo under 60x magnification using a microscope.
This number can then be typed into the GIA website to verify the stones attributes. This number should also correlate to any documentation you have with the stone such as the Laboratory Diamond Dossier (otherwise known as a Certificate). So, why would we need a Laser Inscription? In short, it is for peace of mind for the end consumer, knowing that the stone is the one mentioned in the accompanying report. It is also something that is very difficult to fake. In some cases if a stone is purchased secondhand without a report - reading the report number off of the girdle will be able to tell you the specifications of that particular stone. In most cases the number can be read from a stone even when set, unless of course the number is obscured by a claw or rubover setting.