How does one understand and resolve Zeno's paradox? Zeno, a follower of Parmenides, reasoned that any unit of space or time is infinitely divisible or not.  If they be infinitely divisible, then how does an infinite plurality of parts combine into a finite whole?  And if these units are not infinitely divisible, then calculus wouldn't work because $n$ couldn't tend to infinity.  

Another way to think about it is a flying arrow must first travel half way to the target from where it begins ( the first task), then travel half way to the target from where it is now (the second task), then travel half way to the target (third task), etc... What you get is this...

$$\begin{array}{l}
{d_{Traveled}} = \frac{1}{2}d + \frac{1}{4}d + \frac{1}{8}d + \frac{1}{{16}}d + ...\\
\\
{d_{Traveled}} = d\left( {\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{{16}} + ...} \right)\\
\\
{d_{Traveled}} = d\left( {\frac{1}{\infty }} \right) = 0
\end{array}
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$$

But suppose we wish to calculate the area below a function between $a$ and $b$ say, the bars that compose this area consist of taking a reference point on the first bar $f(a)$, multiply it by $dx$, then using the slope $f'(a)$ as a guide, "jack up" the reference point onto the top of the next bar, multiply by $dx$, jack it up, multiply by $dx$, and repeat this until we reach the final bar (L.H.S.).  The summation of all this yields the exact area. 

So, it's like taking the line segment $ab$ and dividing each piece over and over again.  Per division, the sizes of the pieces are half of what they were before, but there are twice as many of them as before; but as the number of divisions tends to infinity (n tends to infinity), they diminish to almost nothing such that when added back together, they still equal the length of the original line segment $ab$.

How does one understand and resolve Zeno's paradox?
Zeno's paradox is called a paradox exactly because there is a mismatch between a seemingly logical argument that concludes that motion is impossible, and our experience in dealing with reality, which says that there is motion.  

To resolve the paradox, then, you need to figure out where the argument goes wrong. Saying that in your experience motion exists does nothing to get rid of the argument. Indeed, rather than resolving the paradox, when you flap your arms or drop any balls, you emphasize the paradox!

Finally, let me also add that with his argument(s) Zeno most likely wasn't trying to conclude that motion doesn't exist, but instead offered his argument as a reductio ad absurdum against the idea that space is infinitely divisible:  If space is infinitely divisible, then [insert typical Zeno story here] motion becomes impossible. But since [flap your arms now] there is motion, space cannot be infinitely divisible. 

So, you being able to drop a ball to the ground is now part and parcel of the argument!  And if you want to reject the conclusion that space is not infinitely divisible, you need to show how motion is possible in such a space. Dropping balls doesn't demonstrate such a thing, because Zeno will simply say: you were able to drop the ball exactly because space (as in: the real, physical space of the world we live in) is not infinitely divisible.
Consider the following thought experiment, which I will frame in terms that Zeno would most likely have understood.

Take an urn, and fill it with water.  It will contain a finite amount of water.

Now, transfer half the water in the urn to a second urn.

Then transfer half the water in the second urn to a third.

Repeat this process with a purely hypothetical infinite collection of urns.

How much water is present in the infinite collection of urns?  Other than the initial filling of the first urn, at no time was any water added to or removed from this thought experiment.

Is it finite because we know how much was present at the start and that quantity hasn't changed, or is it infinite / otherwise impossible to determine due to being an "infinite sum"?