There are three relativistic effects that occur on objects with mass. These are length contraction, time dilation, and mass increase. Relativistic effects occur on any object with mass moving at any speed, but these effects are infinitesimally insignificant unless the object is moving at a noticeable fraction of the speed of light (approximately 3.0 X 10^8 m/s).
One of these effects, time dilation, has been detected on spacecraft orbiting the Earth. A spacecraft such as the shuttle orbits the Earth fast enough and long enough to slow internal clocks by a few seconds in relation to previously synchronized Earth based clocks. One of the easiest ways to understand this effect is to think of objects with mass as traveling through space-time, instead of just space. All objects with mass travel with a constant "speed" through space-time (It's not really velocity, as velocity is motion over distance, not distance and time). If an object is stationary in relation to its surroundings, it is traveling only through time, at it's constant speed. In fact, the object is traveling through time alone at the speed of light. If the object begins to move, it's speed through space-time remains constant, but it's speed through time must slow down to compensate. If the object moves through space at a speed approaching the speed of light, it's speed through time diminishes significantly. If it moves through space at the speed of light, it will stop "moving" through time altogether.
Lack of time is the least of the object's worries though. As an object's speed approaches the speed of light, it's length gets shorter in the direction that it is traveling. If it travels at the speed of light, it will have no length at all! This is called length contraction.
In addition, an object moving at a speed approaching the speed of light will get more massive. If it moves at the speed of light, it will have an infinite mass. In fact, it would take an infinitely large force to accelerate a mass to the speed of light. This is mass increase.
So you might ask, how does light travel at the speed of light if this is so difficult? The answer is that light does not have mass. Photons, the particles that make up light have energy, but no mass. Particles that make up you, me, and the Earth are up quarks, down quarks, and electrons, which each have varying amounts of mass and energy.