The Newtonian universe, as imperfect as it could be, allowed a lot of progress in astronomy. It remained unchanged during more than two centuries, till the beginning of the XXth, when Einstein proposed the theory of relativity, as an answer to the dead end reached by physics.
The speed of light is finite : the first one to realize this fact was Oleaus Römer in 1675, when he studied the satellites of Jupiter and their eclipses.
The undulatory characteristic of light was put in evidence in the
XVIIIth century, and in 1817, Fresnel proved that
it was a transverse undulatory vibration. If there is a vibration, he thought
that a support was needed : it will be the aether, infinitely stiff, but offering
no resistance to the movements of celestial bodies.
In 1887, Michelson and Morley showed, in a famous experience, that, if this aether was real, then the Earth had no speed with regard to it.
The successive failures of classical mechanics, and its obvious incompatibility with electromagnetism brought Einstein to the theory of special relativity , which leans on two fundamental principles :
Physically, the first principle means that there is no absolute spacetime, no absolute frame of reference with respect to which position and velocity are defined. Only relative positions and velocities between objects are meaningful.
As a result of the second principle, the galilean law of addition of speeds becomes false.
Of course, these two expressions are equivalent, if the speeds are low relatively to the speed of light.
As a consequence, the measures of time, length and energy are relative, i.e. specific for each observer.
Hence, the distance between every couple of points located on a light cone is zero (ds=0), and the only reachable points are those which are inside the cone, because the interval ds² between two points must always be positive.
If you want an easy way to display the relativistic effects, I suggest you to visit this page where you can play with an applet. It will show you the time dilatation and length contraction at relativistic speeds.
Ten years after special relativity, Einstein generalizes his principle of equivalence to all the systems of reference, whatever motion they have. Einstein puts then the principle of equivalence : acceleration and gravitation are indiscernible. It means that you can't find an experiment which could allow to decide whether your system of reference is accelerating - a rocket which takes off for exemple - or is situated in the gravity field of a mass - on the surface of Earth or any other star.
Unlike the absolute Newtonian space, this one is bound to its contents. It is not pre-existent and its geometry will come from the presence of masses. Hence, these masses will modify the behaviour of bodies and light.
The rigid Newtonian universe is so replaced by a four-dimensional Riemann space-time, which is bended by the presence of masses.
What are the main consequences of this theory for astrophysics?
All these effects have been experimentaly measured.
Another experimental test, the first one that has been led, concerns the advance of the perihelion of Mercury. This planet has a very eccentric orbit, hence great variations of speed. General relativity is the only theory able to explain why its perihelion regularly advances of 43 seconds each century, once shielded the influence of other planets.