I'll try and explain relativity in a simple way most people can understand and keep the formulas to a minimum.
Imagine you're riding in a car down the road at 55 miles per hour with a baseball bat in your hands. You're in the passenger's seat, and you see your annoying neighbor's mailbox. You stick the bat out the window and swing. It's going to cause a hell of a lot more damage than if you were standing still, despite the fact that your basic action was the same. This is because of velocity addition and the physics concept of frame of reference.
A frame of reference is simply a perspective. To the passenger's perspective, they were swinging a bat the same as if they were standing on solid ground. To the mailbox's perspective, a bat was being swung at it from a fast moving vehicle. Classical (i.e. Newtonian, pre-relativity physics) held that the speed at which the bat collided with the mailbox is simply the speed of your swing plus the speed of the vehicle. Even though the frame of references are different, the power of the collision is the same, it's simply that from the passenger's perspective the mailbox is moving at him at 55 miles per hour (oriented in the opposite direction of the ground-frame of reference's car movement).
In Newtonian Mechanics (classical physics), frame of reference does not really affect the underlying calculations at all. For the longest time we thought that the Newtonian formulas were perfectly acceptable ways of describing the laws of motion, and there was a reason for that. We only really verified them experimentally within the realm of "basic" motion (i.e. day to day movement). Fast speeds were not taken into account and the speed of light was not something that could be calculated at the time, so no one really worried about it. Here comes along the problem:
Say you're riding in a vehicle and you fashion a mirror to the ceiling directly above you. You place a special sensor around the opening of a flashlight so that it records the time that a single photon leaves the entrance of the flashlight, bounces off the mirror, and returns, basically measuring how long it takes for that photon to reach the ceiling and come back. This censor is perfectly accurate (for the sake of this example) to several bazillion decimal places. Newtonian physics guarantees that no matter what you're doing, the travel time recorded from the flashlight to the ceiling and back should be the same.
The problem is, when you're moving in a car as opposed to being completely stopped, the flashlight does not simply go up at the ceiling and come back down; it's more of a ^ motion, because the mirror and the flashlight have moved in the direction of the road by a microscopic amount since the photon left the flashlight and also gone up and down at the same time. This means that the photon actually travels further by a minute amount while the car is moving. The same can be said of a ball that you throw up and down in the air while the car is moving (except that would move in a perfect upside-down u shape due to the influence of gravity). The only way that the photon can cover a larger distance in the same amount of time is by moving faster (I'll come back to this in a second).
Because the earth itself is moving, both through space and rotating on its axis, if light moves at a uniform speed, it would be difficult to detect what that speed is due to the property of velocity addition. Several scientists in the 19th Century tried to devise incredibly complicated ways of figuring out just what the speed of light is, while also taking into account velocity addition. The Ether Frame, was the name that they gave to the "perfect" frame of reference where light moved its natural speed and no velocity addition was occurring resulting in an accurate measurement. Unfortunately the search for the ether frame failed; no matter how they measured the speed of light, the answer was always the same.
We now know that this is because light always travels the same speed regardless of the frame of reference that it is in; light cannot speed up to account for differences in the frame of reference. Unlike everything else, its speed is not relative to any observational perspective.
This has all sorts of crazy implications. Remember the earlier scenario where I described light moving in a ^ instead of a straight line because of the moving car? Well, if light can't speed up in order to make up for the longer distance, then what does it do? The distance formula holds that distance traveled is equal to the speed at which it travels multiplied by time. Turns out that if light can't increase in velocity, then time is gracious enough to exercise patience. This effect is known as time dilation, and all sorts of crazy outcomes occur as a result.
The equation is: Time Experienced by Person Moving = Time Experienced by Person Not Moving/(1 - (v/c)^2)^(1/2) where v is the velocity difference between them and c is the speed of light. Or to describe it in words, you start by figuring out what fraction of the speed of the light the person is moving at. You square that, subtract it from one, and then take the root. Then you divide the speed the other person is moving at by the result. For example, lets say they're moving at half the speed of light, then time is dilated by a factor of 1.15; so about 15% extra time.
For example: as a result of this, whenever an object possesses a velocity, it moves faster through time than the surrounding stationary objects. The effect isn't very large because the equation involves the speed of light, but at the sort of speeds we see in space, it starts becoming very noticeable. If we ever succeeded in creating a spacecraft capable of reaching distant objects in a reasonable amount of time, time dilation would result in all sorts of crazy things. Imagine two twins, one becomes an astronaut and visits mars. When he returns, he will be significantly older than the twin he was born with due to time dilation. This is known as the Twin Paradox.
Time Dilation has been tested and experimentally verified by use of atomic clocks orbiting the earth as a satellite at a very fast speed, and it turns out that relativity effects nearly every aspect of Newtonian Mechanics. The formula for velocity addition had to be reworked in order to impose a "universal speed limit" that keeps anything from ever reaching the speed of light (except a massless particle such as a photon). Even the formula for momentum had to be changed in order to reflect this unusual, counter-intuitive discovery.
That's the basics, but it expands much farther beyond that when you start talking about how gravitation effects things. I don't feel like I'm capable of explaining that simply because there aren't really a lot of good thought experiments for that. Special relativity refers to time dilation due to velocity, general refers to time dilation due to gravity. Both basically alter the way that those who are undergoing gravitation or velocity experience time.
Tada; this on its own doesn't really affect religion directly, but it is rather mindbending. It's also a great impediment to long-distance space travel. If we ever managed to go .9c, that still wouldn't be fast enough to get us where we're going at a reasonable rate when they places we want to visit are so many lightyears away, and time dilation makes it even worse. At .9c a year becomes 2.25 years relative to the earth; a trip across the stars would take several generations. You'd basically need a habitat capable of high-velocity, high-energy motion. Nevermind the problem of colliding with crap.
EDIT: I too feel dread; that no one will actually read this. I guess I can't blame them.