Light is an electromagnetic wave.  The electric field of a light wave propagating in the z-direction is given by E = E₀exp(i(kz - ωt)).  The time-averaged intensity of the wave is (1/(2μ₀c))|E₀|².  Light is a transverse wave.  E is a vector lying in the plane perpendicular to z,  E = (E_x, E_y).  Polarized light is produced when the direction of E in the plane perpendicular to the direction of propagation is constrained in some fashion.

The electric field vector E can always be resolved into two perpendicular components.  The light is elliptically polarized when the two components have a constant phase difference, and the tip of the electric field vector traces out an ellipse in the plane perpendicular to the direction of propagation. 

E_x = E_0xexp(i(kz - ωt)),  E_y = E_0yexp(i(kz - ωt + φ)). 

Linearly polarized light is a special case of elliptically polarized light.  If the light is linearly polarized, then the two components oscillate in phase, for example
E_x = E_0xexp(i(kz - ωt)), E_y = E_0yexp(i(kz - ωt)), φ = 0. 
The electric vector traces out a straight line in a plane perpendicular to the direction of propagation of the wave. 
For example, E = Ei = E_0xexp(i(kz - ωt))i.

Circularly polarized light is also a special case of elliptically polarized light, in which E_0x = E_0y, and the two components have a 90° phase difference.  The electric field vector then traces out a circle in the plane perpendicular to the direction of propagation.  When viewed looking towards the source, a right circularly polarized beam has a field vector that describes a clockwise circle (φ = -π/2), while left circularly polarized light has a field vector that describes a counter-clockwise circle (φ = π/2) at a fixed position z.

The figure below shows the trace of the field vector E_x = E₀cos(kz - ωt), E_y = E₀cos(kz - ωt + φ) in a plane perpendicular to the z-axis when looking towards the source.  (E_0x = E_0y = E₀).

φ = 0 Ex = cos(ωt) Ey = cos(ωt)	φ = π/4 Ex = cos(ωt) Ey = cos(ωt - π/4)	φ = π/2 Ex = cos(ωt) Ey = cos(ωt - π/2) = sin(ωt)	φ = 3π/4 Ex = cos(ωt) Ey = cos(ωt - 3π/2)
φ = π Ex = cos(ωt) Ey = cos(ωt - π) = -cos(ωt)	φ = 5π/4 Ex = cos(ωt) Ey = cos(ωt - 5π/4)	φ = 3π/2 Ex = cos(ωt) Ey = cos(ωt - 3π/2) = -sin(ωt)	φ = 7π/4 Ex = cos(ωt) Ey = cos(ωt - 7π/4)

You can also open this Excel spreadsheet and explore.
It contains a macro, so you must download it to run it.

• Linearly polarized light:  φ = nπ, n = 0, 1, 2, ...
• Circularly polarized light:  E_0x = E_0y, φ = nπ/2, n = 1, 3, 5, ...
• Elliptically polarized light:  φ = arbitrary, but constant.
• Unpolarized light: φ, E_y, E_x are randomly varying on a timescale that is much shorter than that needed for observation.

Natural light is, in general, unpolarized.

AI Study Tip:
Example Prompt:  'Explain the difference between linear, circular, and elliptical polarization using an analogy involving a person shaking a rope tied to a wall.  How does the 'phase shift' between the horizontal and vertical components translate to the motion of the rope?  Create a table comparing the math (E-field equations) to the physical rope analogy.'