Dark matter is matter that is invisible to our normal methods of observing matter in galaxies, yet it still has gravitational effects on visible matter. No one is quite sure what dark matter consists of and it's a topic of a lot of ongoing research.
Dark matter was originally theorized by an astronomer by the name of Fritz Zwicky who saw a large discrepancy between the observed mass of galaxies (the visible matter) and their computed masses (computed from gravitational effects). This discrepancy is known as the "missing mass problem." This missing mass is termed dark matter -- it is "dark" because it's not visually observable.
Understanding dark matter helps us understand the history of the universe. Our current knowledge of the formation of galaxies is not consistent with theories that do not involve dark matter, so the more we know about dark matter, the more we understand how these galaxies originally formed. Also, it helps us understand current behavior of galaxies. Without dark matter, objects further away from the center of a galaxy should move slower, but experimental evidence shows that instead, this motion is constant after a certain radius. Dark matter explains this anomaly.
An important concept related to dark matter is dark energy. Dark energy helps us understand the universe's expansion. More information on dark matter and dark energy can be found here and here.