Grout is the material that is applied between tiles to fill in the spaces and give strength to the tilework. For tiles that are close together (1/16th inch, or about the thickness of a penny), non-sanded grout is used. For tiles wider apart than this, sanded grout is used.
Grout is applied after the tile is laid down on its bed of mortar. The grout is squeezed into the spaces with a tile float, and then wiped clean away with the same tile float and with a wet sponge.
Grout is more than a utilitarian material. Grout of different tints, textures, and colors can be used to give your tile work completely different looks.