What is Total Internal Reflection?

In short

Total internal reflection happens when light tries to leave a denser medium for a rarer one at an angle larger than the critical angle, and all of it is reflected back inside.

In simple words

Two things must be true. The light must travel from a denser medium to a rarer one, for example glass to air. And its angle of incidence must be larger than the critical angle $\theta_c$ , where $\sin\theta_c = \dfrac{n_2}{n_1}$ . Optical fibres, diamonds' sparkle and a mirage all depend on this effect.

Example

For glass of refractive index $1.5$ in air, $\sin\theta_c = \dfrac{1}{1.5} = 0.667$ , so $\theta_c \approx 42^\circ$ . A ray inside the glass meeting the surface at $50^\circ$ is totally reflected.