As an example of how important: the Global Positioning System (GPS) uses atomic clocks for its timing. These clocks, orbiting in space at ~17,000 mph, are time-dilated due that velocity, that is, their time drifts relative to our time on Earth. Relativistic calculations are used to correct this drift and send the right timings to GPS devices in planes, ships, cars etc.
Depends on what you think of your relatives! They will have more influence on your life than the "Theory of Relativity." Time dilation, change of mass with velocity etc has little effect on the average day to day person. Sure it is a nice bit of trivia to know that the speed of light seems to be an upper limit at which mass may travel...but getting there is going to be a bitch.