A device for counting blood cells, originally, but now used to count any free-floating cell. Variably spelled hemacytometer, haemocytometer, and haemacytometer, though I believe hemocytometer is correct, at least in the USA.
The device is basically a special microscope slide, made to high tolerances. There is also a very flat, reusable cover slip. Basically, the hemocytometer is made such that the thickness between the slide part and the cover slip is constant. And the slide part is etched carefully, so that when one looks at the slide through a microscope, you see squares. Because the height is constant, the volume of fluid in the square is known.
It is used by putting the cover slip on the device, and filling the space with a liquid containing the cells you want to count. The fluid is usually drawn into the space by capillary action. Then one counts a number of squares, and you work out how many cells you counted per volume counted. This gives you the number of cells per volume, and you multiply by the total volume of fluid containing the cells.
There are also versions made for counting much bacteria, which are much smaller. These devices are made such that the height is much smaller than in normal hemocytometers.