Relativistic ship and base on Earth talking Let say humans make it possible to travel so close to the speed of light that time dilation for the ship can achieve 100 times the time of a standing still object. So one of that ships starts its journey and after a while something goes wrong and they have to call the base on Earth. The chat consists of two sentences that the pilot on board says in 1 minute. My question is why his sentences the people on the base should listen for almost two hours? To not have problems with signal shift due to ship relative motion regarding Earth let assume the ship is orbiting Earth at a large but constant distance.
 A: Let's say there are two events:
A. The pilot starts their message
B. The pilot finishes their message
In the rocket frame, which we'll call $S^{\prime}$, the time between those events is $\Delta t_{AB}^{\prime}$ as you've said in the question. To work out the time between the message being first received and ending on Earth, the $S$ frame, we'll need to Lorentz transform.
\begin{equation}
\Delta t_{AB} = \gamma \Delta t_{AB}^{\prime}
\end{equation}
where
\begin{equation}
\begin{split}
\gamma &= \left(1 - \frac{v^{2}}{c^{2}}\right)^{-\frac{1}{2}}
&= 100
\end{split}
\end{equation}
from the information you've given about the relative velocity of the ship and someone standing still on Earth. Therefore $\Delta t_{AB} = 100 $min.
This effect is known as time dilation. Special relativity predicts that, in order for the speed of light to remain constant in all inertial frames, time has to run at different rates for observers moving with different velocities.
EDIT:
As WillO has pointed out in the comments of your original question and both answers given, the ambiguity as to whether the ship is orbiting or travelling towards/away from Earth is actually quite important. The answer given here is based on the assumption that the ship is orbiting Earth, however in the other two cases you must consider the relativistic Doppler effect as detailed in the answer below.
