How to explain space-time curvature in two minutes? How would you explain what is the curvature of space-time in a short period of time like 2 minutes to non experience people? 
 A: Imagine space-time as a flat sheet of elastic fabric.  Put a grapefruit on it, and it will sag and form a depression around the grapefruit.  Now try to roll a marble straight across the fabric. If it passes near the grapefruit, it will start rolling into the depression.  If the marble is rolling fast enough, it will begin to circle the grapefruit before falling into it.  But if the marble is going really fast, it will turn into the depression and then will continue to roll through it and up the other side, although its trajectory will have been bent.  Here is a diagram: http://theory.uwinnipeg.ca/users/gabor/black_holes/slide5.html
This is what happens to objects and even to light moving through space around a massive gravitating body.  Flat space doesn't interfere with trajectories, but curved space does.
Time changes along with space in the presence of massive gravitating bodies.  Notice as the marble circles the grapefruit, it speeds up and circles faster as it gets closer to the bottom of the depression.  In the same way, clocks run faster as they fall deeper into a gravity field.
Curved space-time alters trajectories of objects and of light in the presence of gravitating bodies, and it speeds up clocks as they fall into gravity fields.
Caveat:  This analogy is not the real story.  But it may help, if not taken too literally.  Here are two critiques of the elastic fabric analogy.  Also see rob's answer, a profound and eloquent way to conceive of curvature.
A: http://www.einstein-online.info/spotlights/light_deflection
Tell them about how a light ray from a distant star is deflected as it passes through curved space near the sun, making the star look as though it has moved position. The above link should give you the details of how, without curved space, the  light ray is predicted to be half the angular deflection it actually is. And a diagram always helps, best of luck with it.
A: This started out as a comment on Ernie's answer, but I like it enough to put it here on its own.
Lots of people have heard of the "rubber sheet" analogy: you imagine (or see with your eyes; it's a common demo to build) some masses sitting on a piece of spandex.  The large masses make deep indentations in the spandex, and the small masses make shallow indentations.  The small masses tend to move into the indentations caused by the big masses.
I don't care for this analogy for unequal masses, because the thing that moves the small masses into the deep well is … gravity.  However, if you imagine two large, roughly equal masses near each other on a rubber sheet, you can convince yourself that the shape of the sheet becomes less complicated as the masses move together — the saddle point between them goes away.  That's a curvature force.
You get curvature forces in soap bubbles.  A floating soap bubble is roughly spherical.  But when two soap bubbles stick together, neither bubble remains spherical: the membrane between the bubbles moves so that its curvature is reduced, often to a plane.
A: You can walk halfway around a block two ways - east then north, or north then east. In flat space either path takes you to the same place.
This does not happen in curved space. The surface of the Earth is gently curved, but you can see it if you use a big enough block. On a globe start at the equator. 


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*Go east 1/4 of the way around the world. Turn right and go north to the north pole. 

*Go north 1/4 of the way around the world. Turn left and go to the equator.
You wind up in different places. 


A large mass like the Earth induces distortions in space time. Time runs slower near the earth. See this post for an explanation.
This distortion is a curvature of space time. But in this case, the "directions" around the block are in the timeward and radial directions. You cannot move to another point at the same time, so we will just locate events in space time that form the corners of a rectangle. 
Start at the top of a tall tower at time $t_0$. 


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*Locate the point at the top 1 second later. Then locate the point at the same time at the foot of the tower. 

*Locate the point at the foot of tower at time $t_0$. Then locate the point at the foot 1 second later. 


Since time runs slower at the foot, you wind up a the same point, but at different times. Space time is curved. 
A: Since spacetime is four-dimensional, stars, planets, rocks, and people are four-dimensional. In spacetime, a star is not a ball in the middle of a dimple; it is more like a rope going from the past to the future, with the planets like strings spiraling around it. A boulder that is initially at rest is a string that starts out pointing "straight into the future" and then curves until it touches the rope of the sun. (Notice that we're ignoring one of the spatial dimensions so we can see the time dimension.)
Curvature, in everyday life, is mostly about how lower-dimensional things fit into space. But, from the point of view of a denizen of the curved surface (or curved 4-space), what matters is how it affects distances and angles. On the curved surface of the earth, if two people start out at the equator heading due north, the distance between them will decrease. Since we live within spacetime, all that matters to us about curvature is how the measurements of distance (and time, which is part of the 4D distance) change in different parts of spacetime. You may have heard that time moves at a different speed near massive objects, and distances are actually different too. This is the curvature.
The rule for curvature and motion is this: In free fall, the rope or string of an object (its world line) follows the shortest path through spacetime. So think of the rock that starts out still relative to the star (its string is pointing straight into the future). Think about a point one day in the future, but the same distance from the star. The shortest spacetime trip to that point will veer out away from the star before curving back to the point, just like the shortest plane flight from London to New York veers up to Greenland on a map. If the rock starts out "still", with its string pointing straight into the future, the only possible curve that's a shortest spacetime path to somewhere will curve into the star's rope.
This may seem wrong if you know that time slows down near (say) a black hole, but there is a minus sign that comes in when you combine time distance with space distance to get spacetime distance.
Also, it is possible mathematically to put spacetime in a somewhat higher dimensional space so that there is "no distortion", but there is no reason to think that has any physical meaning. It's the spacetime distance, and more generally what's called the metric, that matters.
