One was about a theory of Special Relativity that explains that the laws of physics are invariant to an observer moving at a constant speed. In the theory the concept of Space-Time as a four-dimensional universe is defined. Most important, Einstein showed that no physical interaction can propagate faster than light.
Later, Einstein realized that this theory was still incomplete because observers could move at variable speed, e.g. under acceleration. Also, there was no account for gravity: the attracting force between the masses of objects. The classic approach according to Isaac Newton, is that gravitational force is acting directly and immediately between objects.
In 1915, Einstein published the theory of General Relativity (GR) that took account of the missing details. Very shortly summarised, the theory extends the concept of Special Relativity and explains that the fabric of Space-Time is locally deformed in the presence of a massive object. The deformation may be observed as what Newton would have called a gravitational force.
Of course, a theory requires some experimental proof. Already Einstein had considered that a massive object would locally curve space-time. Consequently, light travelling through Space-Time in the vicinity of a massive object would follow a curved path (as seen by an external observer). Starlight that passes the Sun closely, would pass through curved Space-Time and would be deflected. The deflection angle corresponding with the mass of the Sun, could be derived from GR theory.
Hundred years ago, Arthur Eddington, a British astronomer and physicist, was aware of Einstein’s publication on GR and was intrigued to try measuring this phenomenon. If the effect was seen, it could prove the GR theory. Since it is virtually impossible to measure starlight near the bright Sun, such measurement could be done on Earth only during a total eclipse of the Sun.
In 1919 a total eclipse of the Sun would take place over a part of Africa and South America. Eddington (and scientist Frank Dyson) organized expeditions to locations on the path of the eclipse to look for the starlight event. Eddington went to Principe Island on the west coast of Africa.
It is all history. On 29 May 1919, Eddington managed to measure the deflection of light of a star while it passed behind the eclipsed Sun. Measurements were in agreement with Einstein’s prediction and confirmed GR for the first time. Worldwide publicity made both Einstein and Eddington famous.
Shortly after World War I, Eddington’s plan to carry out this experiment was far from simple. In retrospect, it was a very remarkable achievement, not only since it confirmed GR for the first time. The expedition’s team had to travel by ship across the globe to the locations of the eclipse and try to measure starlight with the comparatively simple technological means of that time (during the eclipse, a series of photos was taken with a camera on a telescope). Nowadays, eclipse watchers can routinely go and watch/record an eclipse (but still this is an experience).
Relativity has effects in Space-Time by both velocity and gravity. On-board a space ship that travels at high speed, time will pass slower than on Earth. Also, gravity influences the clock rate by slowing it down in regions with large gravity in comparison to regions where there is less gravity. On Earth time passes slower than at a location with less gravity.
In the Apollo project to bring humans to the Moon, NASA concluded that for calculation of the trajectory of an Apollo space ship no relativistic corrections were required. It turns out that Apollo Guidance Computer (AGC) flight software determined the flight trajectory based primarily on Earth’s gravity constant and the Moon’s gravity constant. Space flight to the Moon was still slow and relativistic effects were small.
In contrast, physical events involving more massive objects and/or higher velocity will have stronger interaction with Space-Time. In 2015 an extraordinary gravitational event was detected. A collision of two massive objects at about a billion light-years from Earth was observed by LIGO (the Laser-Interferometer Gravitational-wave Observatory). In the collision (GW150914), two black holes, each about 30 solar masses, merged. The energy released in the merger caused a deformation of the fabric of Space-Time, like a ripple in a pond. The ripple passed our solar system, indeed, at the speed of light. Imagine that.