Lorentz transformations are the heart of
special relativity. In special relativity, time is treated as a fourth dimension in addition to the three space dimensions, time just behaves a bit differently. So in special relativity, many things are represented as four-
vectors, which have a time-component in addition to the three space-components. An example is a four-vector which tells
the position and time when something, or maybe nothing, happens.
Of course, four-vectors are always given _relative_ to somebody or something. For example, if I say: "I see a ping-pong ball 3 m in front of, 1.5 m to the right and 1 m up from my nose" (assume that 'in front of', 'to the right' and 'up' are actual directions!), it is meaningless unless you know where my nose is and where I am facing. If I turn my face 20 degrees to the right, I will have to modify those numbers, in a way that I can work out mathematically! This is a type of a Lorentz transformation.
Now, special relativity extends this to include the time as well! In special relativity, when you change the velocity you are moving at, you rotate in space-time! To return to the ping-pong example, if I say: "1 s from now, there will be a ping-pong ball 5 cm in front of my nose",
if I start moving back at a velocity of 20 m/s, I will have to change both of those numbers _the instant_ I start moving.
Of course, the effects of special relativity can't in practice be detected when playing ping-pong, but in principle they are there. In astronomical scales, special relativity predicts many curious things, like time dilation and Lorentz contraction, all of which have been verified.
Let's get a bit more mathematical. We look at Lorentz transformations with 2 coordinates, t and x, t being the time. We use units of length and time where c, the speed of light is equal to 1. So when we say x=t, it means x=c*t. Now, as you can see, x=t describes the motion of a ray of light along x-axis. Remember, the speed of light must remain the same regardless of the velocity you are moving at. But at first this seems weird, because if you change your velocity from 0 to v, x=t becomes x=(1-v)t by common sense, so after changing your velocity the ray of light would be moving at velocity 1-v. So if the speed of light relative to you must stay the same regardless of your speed, we must extend common sense a bit.
Now, it turns out that Lorentz transformation is a linear transformation, which means that it can be represented with a matrix.
For example, the following matrix equation represents ordinary rotation through an angle phi in a plane:
(x') = (a b) (x) (equation 1)
(y') (-b a) (y)
a = cos(phi)
b = sin(phi)
Note that in rotations the length of a vector, x²+y², stays the same. Actually this, and the fact that a rotation mustn't 'flip' the object that is being rotated, can be taken as the definition of rotation! I should really explain more about this flipping thing some time, because that kind of transformations are of interest in
quantum field theory
Anyway, you can prove using simple trigonometry that x'²+y'² = x²+y² under equation 1.
In Lorentz transformations what stays constant is not t²+x² but rather t²-x²! (You should have no trouble proving that if changing the velocity is expressed by a transformation where t²-x² = constant, it means that the speed of light remains the same.)
This leads (modulo flipping, again) to the following matrix equation:
(x') = gamma * (1 v) (x) (equation 2)
(t') (v 1) (t)
gamma = 1/sqrt(1-v²)
This is a Lorentz transformation you can use when you change your velocity from 0 to v. Again, you can rather easily prove that t'²-x'²= t²-x².
If you want to read more, special relativity is explained more pedagogically for example in the book "Spacetime physics" by John Wheeler and Edwin Taylor.