A combination of uniform circular motions could consist of a large circle (the deferent - Circle DBC with Earth E at its center), carrying around at constant speed a smaller circle (the epicycle), which in turn carried around the Planet (P) at a constant speed.
These 3 images courtesy Norriss S. Hetherington.

The combined movement of epicycles and deferent could reproduce the observed "retrograde" motion of planets. As seen from the Earth in the center at times 1, 2, 3, and 4, the planet apparently moves against the sphere of the stars from 1 to 2, turns back to 3, and then resumes its forward motion to 4. Earth, moving faster than the outer planet, overtakes and passes it.

Besides the epicycle hypothesis, there was also the eccentric hypothesis, in which the large circle was no longer centered on the Earth. It became an "eccentric," centered on a point near the Earth, E. In the eccentric hypothesis, the planet P moves with uniform circular motion along the circle APDB with its center at C. To an observer on the Earth, the motion appears to speed up and slow down.

Here's an animation of the effect: Ptolemy's Model for the planet Mars, by Craig McConnell.
Source: http://faculty.fullerton.edu/cmcconnell/Planets.html#2

As a last resort, the uniform angular speed might be measured not about the center of the circle but about some other point, the equant, A. The Sun, S, moves on the Earth-centered circle, but it does not move at a uniform rate. Its rate of motion is set by the condition that the angle a varies uniformly with time.
Adapted from Thomas S. Kuhn, The Copernican Revolution (1957), p. 71.

The outer sphere is not shown in the drawing. Its axis of rotation is vertical, in the plane of the screen, in a north-south direction. The outer sphere carries everything within it eastward. The axis of the inner sphere is horizontal and in the plane of the screen. The motion of a planet carried about by the inner sphere is up (north) and down (south) and into (west) and out of (east) the plane of the screen. The planet appears to move north and west from 1 to 2, north and east to 3, south and east to 4, and south and west back to 1. When the inner sphere is imparting an eastward motion to the planet, moving the planet from 2 to 3 to 4, the total eastward motion, including the steady eastward motion imparted by the outer sphere, will be very rapid. If the westward speed imparted by the inner sphere is greater than the steady eastward motion imparted by the outer sphere, then the planet will appear to slow down and briefly move west during the passage from 4 to 1 to 2, when the westward velocity of the inner sphere is greater than the eastward velocity of the outer sphere.
Image after N. Hetherington, Ancient Astronomy and Civilization (1987).

Though detailed geometrical models were generated primarily using Platonic two-dimensional circles, it was also possible, at least in principle, to account for such phenomena as the observed retrograde motion of the planets with Aristotelian three-dimensional spheres.

Arab astronomers over later centuries continued to refine the Greek schemes. They introduced still more ways to account for the observed planetary motions. To appreciate their ingenuity, see these links to further technical explanations (and animations).