When an object moves in a circle at a constant speed, its direction constantly changes. A change in direction causes a change in velocity. This is because velocity is a vector quantity - it has an associated direction as well as a magnitude. A change in velocity results in acceleration, so an object moving in a circle is accelerating even though its speed may be constant.
Gravitational attraction provides the centripetal force needed to keep planets in orbit around the Sun and all types of satellite in orbit around the Earth. The Earth's gravity keeps the Moon orbiting us. It keeps changing the direction of the Moon's velocity. This means gravity makes the Moon accelerate all the time, even though its speed remains constant.