CIELab is the color space that ICC Profiles and CMMs often use as an intermediary space when converting colors. So a monitor to printer match translates colors from the monitor's space (RGB) into Lab and then into the printer's color space (CMYK for example).
The L component is the lightness of the color. The a component is the red/green scale (+a is red, -a is green) The b component is the blue/yellow scale (+b is yellow, -b is blue... go figure)
CIELab, or more correctly CIEL*a*b* is a (mostly) device independent color-space based on the measurements of hundreds of humans the CIE made in 1931 when they created the CIEXYZ color space. In 1976 the CIE created the Lab space to reflect the entire gamut or range of colors the human eye can typically see. The Lab space, unlike other CIE color spaces, is supposed to be perceptually uniform. That is, any movement within the space, in any direction, should result in an equally perceptible color shift. There are many who believe that the Lab space is not perceptually uniform but that is outside the scope of this glossary.
In order to define precise points in CIELab which can be encoded into 8 or 16 bit for a PCS, the ICC constrains the a and b axis to 256 points. The L axis is usually bounded by 0 and 100. However the a and b axis can go out as far as is needed.
To view the actual human perceptible gamut, see Bruce Lindbloom's Lab Gamut Display.
LCH is another way of measuring the same color space. (see LCH for more information) The Lab color space is not precisely device independant as it is defined relative to a reference white point. This white point is often based on the whitest point that can be generated by a device or a standard white point (like D50).