Overview
A Lorentz transformation describes how the coordinates of one reference frame can be computed from those of another, given the relative velocity of the two observers.
Interval
Given two points {% (x_1,y_1,z_1,t_1) %} and {% (x_2,y_2,z_2,t_2) %} which are connected by the propagation of a photon of light, the following equation holds (where {% c %} is the speed of light)
{% (x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2 - c^2(t_1 - t_2)^2 = 0 %}
By the second postulate of relativity, this equation must also hold in
{% s^2 = (x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2 - c^2(t_1 - t_2)^2 %}
Lorentz Transformation
the simple form of the Lorentz transformations can be written as the following, where we assume that one inertial observer is moving at a constant speed along the x-axis of another observer, and the Cartesian coordinate systems of both observers are aligned.
{% x' = \gamma(v)(x + vt) %}
{% t_1' = \gamma(v)(t + \frac{v}{c^2}x) %}
{% \gamma(v) = \frac{1}{\sqrt{1 - (v/c)^2}} %}
{% x'^{\mu} = \Lambda ^{\mu}_{\nu} x^{\nu} %}
In addition, if we are assuming the normal 3-space, we have that {% y'=y %} and {% z' = z %}.