The theory of Relativity is broken into 2 distinct parts. The Special theory and the general theory. Though mathematically Special Relativity is much easier to understand than conceptually. General relativity is much easier to visualize, but much harder to get mathematically.
Special relativity all deals with the behavior of objects when they approach the speed of light from different frames of reference. The usual frames of reference are the observer and the person or object travelling at near light speeds. This leads to a couple of interesting phenomenon. Time dilation and length contraction.
Time dilation occurs when a person reaches near light speeds, the observer (stationary) will notice the with the person travelling time appears to slow down and stop as they're in a rocket flying through space the closer they approach the speed of light.
However to the moving observer the opposite will have happened, they'll observe time passing as normal, but depending on the distance they travel and come back, the stationary observer will be older than the moving observer due to the effects of time dilation. The person moving at near light speeds doesn't notice the slow down in the effect of time on them. The stationary observer goes through time at the normal speed.
This is commonly referred to as the twin paradox. 1 twin travels for several light years in a rocket at near the speed of light and comes back to greet his stationary twin, but notices that his stationary twin has aged much more than he has due to the effects of time dilation.
Length contraction is also another phenomenon of special relativity. As a moving object approaches light speed it's length appears to contract to the stationary observer. However to the moving observer the distance of objects in front of them seem to contract and they notice none of the effects of the length contraction.
General relativity is the theory about how gravity affects space and time. It predicts strange objects such as neutron stars, black holes, worm holes, etc. The best way to visualize it would be to imagine space and time as a soft sheet of rubber. The stars, galaxies, planets, etc. create dips in the sheet due to their gravitational influence.
To explain the orbits of planets General Relativity states that they orbit the sun like the sun is sunk into the sheet at a fair depth depending on how strong the sun's gravity is. The planets orbits are like rolling a marble along the side of an upright bucket, they're moving fast enough to keep their momentum going and not move up or down along the walls of the divot too far.
If they start to move down the walls that means the sun is drawing them closer and they're losing their orbital momentum, but if they start to move up the sides they're gaining orbital momentum and may escape the sun's pull and go flying off into space.
It also predicts the behavior or stars in certain circumstances, such as stars the collapse and become neutron stars and possibly black holes. It also shows how light is affected by gravity, and that gravity can visibly bend light if it's strong enough.
Light is said to travel along space and time like a golf ball along a green, if it goes near a gravity well it'll dip along the side and right itself again or change it's angle. Much like a golfball that runs just at the edge of the cup on the green but doesn't go in, how it diverts it's straight path a bit.
I realize this is a long explanation, but it's the shortest one I could come up with and still manage to cover most of the bases for both theories. I hope this helps.