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Circular polarization occurs when two orthogonal linear polarization components of equal amplitude differ in phase by 90°, producing an electric field vector that rotates uniformly. Elliptical polarization is the general case with unequal amplitudes or arbitrary phase differences. Quarter-wave plates convert linear to circular polarization.
To understand circular polarization, start from what you already know about wave plates. A quarter-wave plate introduces a 90° phase retardation between the component of the electric field along its fast axis and the component along its slow axis. When linearly polarized light enters a quarter-wave plate at 45° to those axes, the two components start with equal amplitudes and zero relative phase. The plate adds a quarter-wavelength of extra path to one component, so they emerge with equal amplitudes but now 90° out of phase.
What does it look like to add two equal-amplitude, 90°-out-of-phase oscillations at right angles? At time zero, the x-component is at its maximum and the y-component is zero. A quarter-cycle later, x has fallen to zero and y has risen to its maximum. Half a cycle in, x is at its negative maximum and y is back to zero. The electric field tip traces a circle — it rotates in space as the wave propagates. This is circular polarization: neither component dominates, and the field magnitude stays constant while its direction sweeps continuously around.
Elliptical polarization is the general case that brackets all polarization states. If the two components have unequal amplitudes, the field tip traces an ellipse rather than a circle — spending more time in the direction of the larger component. If the phase difference is something other than 90° (but still not 0° or 180°, which give linear polarization), the ellipse is tilted relative to the axes. Circular polarization and linear polarization are special cases of elliptical: circular is the equal-amplitude, 90°-phase case, and linear is the zero-phase-difference limit where the ellipse degenerates into a line.
The handedness of circular polarization — left-circular vs right-circular — depends on which component leads in phase. Right-circular polarization is typically defined as the field rotating counterclockwise when viewed looking toward the source. This distinction matters in optics because many biological molecules interact differently with left- and right-circular light (circular dichroism spectroscopy), and in antenna engineering because satellite signals are often circularly polarized to remain unaffected by antenna orientation during rotation.