The best known of Zeno's various paradoxes is Achilles and the tortoise. It goes thusly:
A turtle starts walking. After giving it a bit of a head start, Achilles shoots an arrow after it. The turtle moves an foot. The arrow moves twenty feet. The arrow will be impaling the turtle shortly... Or will it?
The arrow flies the last twenty inches towards its target, but while it is traversing those twenty inches, the tortoise moves one inch. As the arrow is barreling its way through the last few millimeters, the turtle is lumbering along. The arrow goes one millimeter, the turtle goes one twentieth of a millimeter. Yes, the arrow is getting closer, but as long as the turtle keeps moving the arrow will still have (in this poorly estimated example) one twentieth of that distance to go, and in the time that it takes the arrow to cross that distance, the turtle will have moved just a little farther on. The arrow can never quite catch up.
Well, obviously it can, but Zeno couldn't figure out how it logically could. Hence the paradox: we could think up this logical argument that seemed perfectly correct, but at the same time, we can see that moving things do get hit by arrows, all the time. A most ingenious paradox.
It turns out that a infinite series can add up to a finite number. Someone who knows more about the mathematics involved will have to post the formula.