A dumby like me can help dumb down the explanation for you! Basically, one of the problems facing physics around the turn of the 19th/20th century (whichever 1900 was) was how to explain the invariance of maxwells equations. Simply, the laws of electromagneticism said that the speed of light is always 300,000,000 meters per second. The problem with this is simple enough. If you ever driven a car at say, 50mph and shot a gun that fired a bullet at 500mph, you would realize that the bullet, from a person stationary watching the car go by, would see the bullet go 50+500 = 550 miles per hour. The problem here is that this does not work with light. If you traveled say, 100,000 meters per second down a road and sent off a beam of light, you would expect a stationary person to see it go 300,100,000 meters per second but this in fact, is not the case! Such a result disagrees with the idea that the speed of light is always 300,000,000 meters per second.
The way that this was solved was by linking time and space using the "Lorentz Transformations". Simply, they are equations that make velocities never add up to be over the speed of light which is the "speed limit" to the universe. This has to be the case because a famous experiment called the Michelson-Morley experiment showed with good enough accuracy that the speed of light had to be constant.
One consequence of the lorentz transformations is that when you move relative to someone, time slows down for you! The twin paradox is the effects of "Time dilation", which is time slowing down for someone. If you travel near the speed of light, 1 minute for you in a spaceship can be 20 minutes on earth! So you would age 1 minute while people on Earth age 20 minutes.
The equations for time dilation only allow time dilation to become noticable at speeds nearing the speed of light. At low, everyday speeds however, the time dilation becomes insignificant so we simply use everyday Newtonian mechanics.
