What is rotation curves?

The rotation curve of a galaxy is the variation in the orbital circular velocity of stars or gas clouds at different distances from the center. The manner in which the velocities vary with radius reflects the distribution of mass in the galaxy:
A solid disk would rotate such that the velocity increases linearly with radius (velocity proportional to radius.
In a galaxy in which most of the mass is concentrated at the center, the velocity decreases with the square root of the radius. Since this is the case in the Solar System, where the planets follow Kepler's laws, this behavior is known as "Keplerian decline".
A flat rotation curve, that is, one in which the velocity is constant over some range of radii, implies that the mass is still increasing linearly with radius.
Most galaxies have rotation curves that show solid body rotation in the very center, following by a slowly rising or constant velocity rotation in the outer parts. Very few galaxies show any evidence for Keplerian decline. A flat rotation curve implies that the mass continues to increase linearly with radius. Therefore, the rotation curves of galaxies present strong evidence for dark matter.