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Wave plates (retarders) introduce a phase difference between orthogonal polarization components through birefringence. A quarter-wave plate (retardance λ/4) converts linear polarization to circular; a half-wave plate (retardance λ/2) rotates linear polarization by twice the plate orientation angle.
From your study of optical path length, you know that the phase accumulated by a light wave as it travels through a medium is proportional to n × d — the refractive index times the distance traveled. In a birefringent material, the refractive index is not the same in every direction: light polarized along the crystal's "slow axis" (higher n) travels slower and accumulates more phase than light polarized along the "fast axis" (lower n). A wave plate is simply a thin slab of birefringent material, cut so that the two axes lie in the plane of the plate and light passes straight through. By choosing the plate's thickness d, the manufacturer controls exactly how much phase difference — the retardance — accumulates between the two orthogonal polarization components.
To understand what a quarter-wave plate (QWP) does, think of polarized light as two oscillating components: one along the slow axis, one along the fast axis. If the incoming light is linearly polarized at 45° to both axes, the two components start in phase and with equal amplitude. After passing through a QWP, one component has been delayed by exactly λ/4 relative to the other — a 90° phase shift. Two sinusoidal oscillations of equal amplitude with a 90° phase difference produce circular polarization: the electric field vector rotates steadily at the light's frequency, tracing a helix through space. If the input is polarized at 45° to the QWP axes, the output is circularly polarized. At any other input angle, the output is generally elliptically polarized. Reverse the process — send circularly polarized light back through the same QWP — and it converts back to linear.
A half-wave plate (HWP) introduces a retardance of λ/2 — a 180° phase shift between the two axes. The effect is to flip the sign of one polarization component while leaving the other unchanged. Geometrically, this is equivalent to reflecting the polarization direction across the plate's fast axis — which means the polarization direction is rotated by twice the angle between the input polarization and the fast axis. If you orient the HWP so its fast axis is at 22.5° to the incoming linear polarization, the output is rotated by 2 × 22.5° = 45°. This makes the HWP a versatile polarization rotator: simply by rotating the plate, you can continuously rotate the plane of linear polarization without any absorption loss.
Wave plates are indispensable in optical experiments and instruments. QWPs are used to convert between linear and circular polarization in laser systems, CD/DVD drives (to prevent reflected light from re-entering the laser), and ellipsometers that characterize thin films. HWPs are used to rotate polarization to the preferred angle of a polarizing beamsplitter, to adjust the ratio of power split between two paths, and to compensate for unwanted polarization rotations in optical systems. The beauty of wave plates is that they achieve all of this purely through phase manipulation — no light is absorbed or reflected, and in principle a wave plate can operate at 100% efficiency.