Quote by Tantalos
Till now I understand, the outcome was that light travels in both direction with the same speed, but Michelson made the calculation for the case there was a fixed ether.

MMX was not trying to measure the time or speed of light traveling in different directions. It turns out, although no one knew it at the time, that that kind of measurement is impossible to make. They thought they couldn't make that measurement because they did not have the technology available to make it so they made a different measurement. What they were measuring was the difference in the time for light to make two roundtrips at ninety degrees apart. They thought that if one of the roundtrips was aligned with the motion through the ether, then it would take a different amount of time than the roundtrip at ninety degrees to it.
Quote by Tantalos
But I don't understand the connection between this experiment and the time dilatation.

There isn't a connection. It is only length contraction that they employed to explain the null result of the experiment. Time dilation came later as a necessary feature to go along with length contraction.
Quote by Tantalos
The Lorentz transformation transforms spacetime coordinates of an event that has noticed one observer to the ones that has noticed another observer moving with a different speed.

It can be used for that but it isn't limited to that. The Lorentz transformation more generally transforms the coordinates of any and all events in one Frame of Reference to the corresponding coordinates for the same events in another Frame of Reference moving with respect to the first one. It doesn't matter if there is an observer at rest in the first one or if there is an observer at rest in the second one.
Quote by Tantalos
But after the Spanish Wikipedia http://es.wikipedia.org/wiki/Transfo...B3n_de_Lorentz there is a condition on when to use it:
"Las transformaciones de Lorentz dicen que si el sistema O está en movimiento uniforme a velocidad V a lo largo del eje X del sistema O y en el instante inicial (t = t = 0) el origen de coordenadas de ambos sistemas coinciden, entonces las coordenadas atribuidas por los dos observadores están relacionadas por las siguientes expresiones:"
"The Lorentz transformation says that if a system O is in uniform movement with speed V along the X axis of the system O and at initial time (t = t = 0) the origins of the two systems are also at the same place, then the coordinates of an event are related with:" (the Lorentz formulas).

This is describing the standard configuration for using the Lorentz transformation, which is the one most commonly used, but other forms of it can be used in more complicated situations.
Quote by Tantalos
In other words, as it pertains to the twin paradox: The travelling twin must never change the speed and direction of his movement, otherwise his age calculation will not be valid.
Can someone explain?

Like I said, you don't have to assume that one twin is stationary in the first frame and the other twin is stationary in the second frame, but if you do, then they can never reunite and there will never be any way to compare their ages.
But you can pick the frame in which just one twin is at rest and the other twin is traveling, not in his own frame, but in the frame of the atrest twin. The you can use the formula that Einstein worked out in section 4 of his
1905 paper introducing Special Relativity which looks like this:
τ=t√(1v
^{2}/c
^{2})
where τ, tau, represents the time dilation of the traveling twin and t is the normal time for the atrest twin.
So if you consider the traveling twin to always be traveling at speed v in the rest frame of the other twin, in other words, from the time he leaves until he gets back, he is always traveling at "v", although his direction can be changing, the ratio of their accumulated ages is simply √(1v
^{2}/c
^{2}).