Maxwell's equations tell us how electric and magnetic fields behave at boundaries between different media.  They yield the boundary conditions for the components of E and B normal and tangential to the boundary.  The unit vectors normal and tangential to the boundary between medium 1 and medium 2 are defined in the figure on the right. 
At boundaries between two non-magnetic dielectric materials the boundary conditions are

(ε₂E₂ - ε₁E₁)·n₂ = 0,    (B₂ - B₁)·n₂ = 0,
(E₂ - E₁)·t = 0,    (B₂ - B₁)·t = 0.

For waves propagating across a dielectric-dielectric boundary or from a dielectric into a conductor we use that the tangential components of E and B are continuous across the boundary to derive the laws of reflection and refraction and the ratio of the reflected and transmitted intensities to the incident intensity.

Reflection

Reflection is the abrupt change in the direction of propagation of a wave that strikes the boundary between two different media.  At least some part of the incoming wave remains in the same medium.  Assume the incoming light ray makes an angle θ_i with the normal of a plane tangent to the boundary.  Then the reflected ray makes an angle θ_r with this normal and lies in the same plane as the incident ray and the normal.

Law of reflection:  θ_i = θ_r

Specular reflection occurs at smooth, plane boundaries.  Then the plane tangent to the boundary is the boundary itself.  Reflection at rough, irregular boundaries is diffuse reflection.  The smooth surface of a mirror reflects light specularly, while the rough surface of a wall reflects light diffusely.

Refraction

Refraction is the change in direction of propagation of a wave when the wave passes from one medium into another, and changes its speed.  Light waves are refracted when crossing the boundary from one transparent medium into another because the speed of light is different in different media. 

Snell's law, or the law of refraction: n_isinθ_i = n_tsinθ_t.

θ_r, and θ_t are the angles the incident and transmitted wave vectors make with the normal to the boundary.

When light passes from one transparent medium to another, the rays are bent toward the surface normal if the speed of light is smaller in the second medium than in the first.  The rays are bent away from this normal if the speed of light in the second medium is greater than in the first.  The picture on the right shows a light wave incident on a slab of glass.

One part of the wave is reflected, and another part is refracted as it passes into the glass.  The rays are bent towards the normal.  At the second interface from glass into air the light passing into the air is refracted again.  The rays are now bent away from the normal.

Reflectance and transmittance

At a boundary between two transparent media, light is partially reflected and partially refracted.  The ratio of the reflected intensity to the incident intensity is called the reflectance R and the ratio of the transmitted intensity to the incident intensity is called the transmittance T.  Energy conservation requires that R + T = 1 (if there is no absorption).

R and T depend on the indices of refraction of the two media n₁ and n₂, the angle of incidence θ₁, and the polarization of the incident light.  We distinguish between p-polarization and s-polarization.  Let the plane of incidence contain the normal to the boundary and the incident wave vector k₁.  The electric field vector E₁ is perpendicular to k₁.  If we choose our coordinate system as shown on the right, then plane of incidence is the xz-plane and E₁ may be written as E₁= E_p+ E_s.  E_p lies in the xz-plane and E_s is perpendicular to the xz-plane, i.e. it points in the ±y-direction.  The electric field of the incident light is a linear superposition of p- and s-polarized fields.

For p-polarized light we have R = |r_12p|², where r_12p is the Fresnel reflection coefficient for p-polarization.  We have  
r_12p = tan(θ₁ - θ₂)/tan(θ₁ + θ₂).

For s-polarized light we have R = |r_12s|², where r_12s is the Fresnel reflection coefficient for s-polarization.  We have  
r_12s = sin(θ₁ - θ₂)/sin(θ₁ + θ₂).

For an interactive graph of the reflectance R for s- and p-polarized light as a function of n₁, n₂, and θ₁, click on the figure below to download the Excel spreadsheet.

If θ₁ + θ₂ = π/2,then tan(θ₁ + θ₂) = infinite and r_12p = 0.  If light is reflected, it will have s-polarization. 
The incident angle at which this happens is called the Brewster angle θ_B.  
We then have
n₁sinθ_B = n₂sin((π/2) - θ_B) = n₂cosθ_B,  tanθ_B = n₂/n₁.
Polarized light can thus be obtained via reflection.

The subject of geometrical optics starts with the laws of refraction and reflection for transparent media.  These laws are used to discover the properties of various optical systems, which may contain any number of curved refracting and reflecting surfaces.  In principle one could solve any optical problem by the exact application of the basic laws.  However, it is possible to derive some very useful general results by using approximate forms of the laws, which treat only rays, called paraxial rays, which make small angles with respect to an optical axis.  Often the treatment is confined to spherical surfaces.  This approximate form of the basic laws as applied to paraxial rays in an optical system consisting of spherical surfaces is called the first order theory and constitutes a major part of the subject of geometrical optics.