With only a limited knowledge of general relativity, I usually explain space-time curvature (to myself and others) thus:
"If you throw a ball, it will move along a parabola. Initially its vertical speed will be high, then it will slow down, and then speed up again as it approaches the ground.
"In reality, the ball in moving in a straight line at constant velocity, but the space-time curvature created by the Earth's gravitation makes it appear as if the ball is moving in a curved line at varying velocity. Thus the curvature of space-time is very much visible."
Is this an accurate description, or is it complete nonsense?
Answer
Yes, that's a fair description of what happens though of course from the ball's perspective it isn't moving - the rest of the universe is moving around it.
However statements like this, while true, give little feel for what's going on. Actually it's extraordinarily difficult to get an intuitive feel for the way spacetime curvature works (or at least I find it so!). The notorious rubber sheet analogy gives a fair description of the effect of spatial curvature, but neglects the curvature in the time coordinate and the time curvature is usually dominant since $dt$ gets multiplied by $c$ in the metric.
The motion of the ball is described by the geodesic equation, but a quick glance at the article I've linked will be enough to persuade you this is not an approach for the non-nerd. I have never seen an intuitive description of how the geodesic equation predicts the motion of a thrown ball.
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