If we assume that all the matter that is only visible in star systems (baryonic matter), then we are not able to account for how galaxies rotate. Similarly, we have problems trying to calculate movement in a cluster of galaxies, or the rate of expansion of the universe. It seems as if there is more gravitational interaction happening than just the matter we see. So, we postulate something called "dark matter" that is essentially matter which exerts gravitational force but is not visible to us since we can't see it (any form of electromagnetic radiation). Other observations such as gravitational lensing (light bending around a heavy object) due to invisible/dark objects also indicate the existence of dark matter.
As for you question of why this mass is not to be found in black holes... If this mass were in a black hole, it would be highly localized. Also, (super)massive black holes are typically found only in the centre of galaxies, whereas to explain the observed effects, we need dark matter to be quite well diffused all around the galaxy and some in between galaxies. We take into account gravitational interactions due to the mass which such black holes can have and ascribe the remaining discrepancy to dark matter.
As always, the Wikipedia page on dark matter gives more details if you're interested.