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At Brewster's angle θB = arctan(n₂/n₁), light reflected from a dielectric interface is completely polarized perpendicular to the plane of incidence (s-polarized). Light in the plane of incidence (p-polarized) is fully transmitted, with no reflection. This effect is exploited to eliminate reflections using polarizers at Brewster's angle.
From Snell's law — your prerequisite — you know that when light crosses from one medium to another with different refractive indices, both a reflected ray and a refracted ray are produced, and the angles are governed by n₁sin(θ₁) = n₂sin(θ₂). What Brewster's angle adds is a special geometric condition: there exists a specific angle of incidence where the reflected and refracted rays are exactly perpendicular to each other, separated by 90°. At that angle, something remarkable happens to polarization.
To understand why, think about how electromagnetic waves work. Light is a transverse wave: the electric field oscillates perpendicular to the direction of propagation. The p-polarization component is the part of the electric field oscillating in the plane of incidence (the plane containing the incoming ray and the surface normal). The s-polarization component oscillates perpendicular to that plane. When the reflected and refracted rays are at 90° to each other, the oscillating dipoles that would re-radiate the p-component into the reflected direction cannot do so — a dipole doesn't radiate along its own axis. So all of the p-polarized light passes through, and only s-polarized light is reflected.
The formula θB = arctan(n₂/n₁) comes directly from combining Snell's law with this 90° condition. If θB + θᵣ = 90° (reflected and refracted rays perpendicular), and Snell's law says n₁sin(θB) = n₂sin(θᵣ) = n₂sin(90° − θB) = n₂cos(θB), then dividing both sides gives tan(θB) = n₂/n₁. For light going from air (n₁ ≈ 1) into glass (n₂ ≈ 1.5), Brewster's angle is arctan(1.5) ≈ 56°. The reflected beam at that angle is 100% s-polarized — completely linearly polarized.
This effect has practical consequences you can observe directly. Glare from a wet road or the surface of water is predominantly s-polarized (horizontal). Polarized sunglasses work by blocking s-polarized light, which is why they dramatically cut glare from horizontal surfaces. Photographers use polarizing filters to suppress reflections from windows and water, making them transparent or revealing what lies beneath. Laser systems use Brewster windows — glass plates tilted at Brewster's angle — so that p-polarized light passes through with zero reflection loss, avoiding the ~4% loss that would occur at each surface at normal incidence.