I remember when I was first introduced to this paradox in 8th grade; it was utterly puzzling (until I heard a solution). A simplified version goes something like this:
Achilles and the tortoise are having a 1000 paces race, but the tortoise has a head start of 800 paces (Achilles is much faster than the tortoise). As the tortoise inches forward, Achilles makes a plan.
First, he'll get to the halfway point (let's call it Bob) between himself and the tortoise. Of course, to get to Bob, Achilles realizes, he has to get to the halfway point between himself and Bob (let's call it Jane). And before he can consider anything else, he must first get to the halfway point between himself and Jane (called Sam). [...]
As Achilles continues to think through his plan, he realizes he can never even catch up to the tortoise, let alone win the race. When viewed abstractly, the problem seems to show that nothing can ever pass anything else—that motion is an illusion. Until the early 20th century, there wasn't really a way to handle this paradox appropriately. However, with the advent of infinite series, we can say that the reason Achilles does pass the tortoise is because if you add up all the little pieces (1/2 + 1/4 + 1/8 + 1/16 ...) you get 1 (which represents the total distance between Achilles and the tortoise).
Fun fact: Zeno's paradoxes are considered some of the earliest examples of reductio ad absurdum, also known as proof by contradiction.