Short answers, pulled from the story.
Albert Einstein published his theory of special relativity in 1905 and showed that time dilation concerns the nature of time itself. He was also the first to point out its reciprocity or symmetry between moving frames.
After six months aboard the International Space Station orbiting Earth at about 7,700 meters per second, an astronaut ages approximately 0.005 seconds less than someone on Earth. Cosmonauts Sergei Krikalev and Sergey Avdeev experienced time dilation of about 20 milliseconds compared to time passing on Earth.
Richard Feynman suggested in a lecture that due to gravitational time dilation, the core of the Earth is younger than the crust. Subsequent calculations suggest the core is 2.5 years younger out of 4.5 billion years.
Rossi and Hall published a comparison in 1941 showing cosmic-ray-produced muons traveling at 98% of light speed live five times longer than slow muons produced in laboratories. The lifetime of particles produced in particle accelerators extends due to time dilation.
In-orbit clocks correct for both velocity and gravitational relativistic time dilation effects so they run at the same rate as clocks on Earth's surface. GPS satellites orbit higher where reduced gravity speeds time up more than motion slows it down.