[tintinxx (OP)]

I understand that birefringent material has two refractive indices. How does the direction of light's electric field comes into play? []

[lyner]

The refractive index of a material is a measure of how easily an electric field can make the electrons move from side to side. This affects how quickly the light can propagate through the medium - getting the electrons moving delays the passage of the wave and it goes at less than c. Most substances are isotropic - the refractive index is the same on all axes so the speed is the same for all polarisations.. Some crystals have a large difference in the way the electrons move along one axis and another. If the E field of the light is aligned one way it will travel faster than if it is aligned the other way.
(The direction of the E field is what we call the polarisation.)

There are two visible effects. The two different refractive indices will split a beam of light into two paths when it enters at an angle. You will see two images through a crystal of calcite, for instance, if you get the angles right.
The other effect is the phase difference between the vertical and horizontal polarised components of a light beam.  This effect can be used in a thin sheet  to produce a 'polariser', which will only let one polarization through - as in polaroid specs. LCDs use polarisers, too.

[tintinxx (OP)]

Thank you! That was very helpful! ^^