What exactly is the paradox in Zeno's paradox? I have known about Zeno's paradox for some time now, but I have never really understood what exactly  the paradox is. People always seem to have different explainations.

From wikipedia:

In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 metres, for example. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 metres, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 metres. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise.
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And we then say that this is a paradox since he should be able to reach the tortoise in finite time? For me it seems like that in the paradox we are slowing down time proportionally. Aren't we then already using the fact that the sum of those "time sequences" make up finite time? I feel like there is some kind of circular logic involved here. 

What exactly is the paradox?
It's only a paradox if you assume that the sum of (countable) infinitely many numbers cannot be finite. But modern mathematics has no problem with infinite sums that yield finite results - in the case of Zeno's paradox, the sum in question is $$
  \sum_{k=1}^\infty 2^{-k} = 1 \text{.}
$$

Not everything that is called a paradox is actually a logical inconsistency. Quite often, things only seem inconsistent because we inadvertedly make an additional assumption, which turns out to be wrong. In the case of Zeno's paradox, that is the assumption that infinite sums cannot yield finite results.
I think some people just can't believe that our universe works in such a way that there is an infinite descending chain of times even though the laws of Physics say that time is continuous and therefore there is an infinite descending sequence of times. They might be like "If the universe did work in that way, then some events at some times would have no original cause?" Although Calculus gives a solution to the problem, they might still be like "Of course calculus gives a solution in an ideal mathematical world but how is it possible that our universe really works in such a way that there is an infinite descending chain of times?" You can't mathematically prove that you won't some day go through a tunnel into an entirely different world like in the movie "Coraline", yet you strongly believe it will never happen and would consider it magic if it did happen. Maybe some people feel the same way about the universe working in such a way that time is continuous. Some people might try to come up with a possible solution that the fundamental laws are a Conway's game of life and when a small amount of time goes by in the emulated universe, a much larger amount of time goes by in the Conway's game of life.  There probably is a way for a Conway's game of life to simulate a universe where it's consistent with observations that time is completely continuous. In such a simulation, maybe we can never make a measurement to infinite precision after a finite amount of time but as more time goes by, we can keep making more accurate measurements and that's why we observe time as being continuous even though in the Conway's game of life, it's discrete.