Is time sped up when orbiting a black hole? Why? What does that mean? I was watching a very interesting short documentary in which the author said that while it takes 8 minutes in space to loop around the black hole, an observer from Earth observes that it takes 16 minutes from Earth.
I understand that this has to do with relativity, but does this mean that the astronaut feels that the time passed is 8 minutes while those on Earth feel it took 16 minutes? Or is time actually sped up?
If time is sped up does that mean that doing a chemical reaction takes less time to complete in space, or is it just the same time felt differently by an astronaut? 
 A: It is not space itself that affects time; it is high speed or acceleration/gravity.
And the effect even makes intuitive sense on earth, although the consequence of it is hard to grasp.
Think of you sitting in a train wagon with a light bulb. Turn it on, and the light starts flowing up to the ceiling where you have a detector. You can measure the time it takes for light to move this vertical height. 
But now imagine that the train is moving. You still see the same thing. But a person on the platform outside sees the light bulb being turned on when you are at one point and before the light reaches the ceiling the trains moves to another point. The starting and end points of the light are not vertically above each other. The endpoint is skewed sideways.
See some of the illustrations here to visualize it.
The distance between start and the end point is now "larger" than if the train was standing still due to simple geometry. But you don't know that inside the train. You measure the same time duration. The guy on the platform measures another time duration because the light is moving farther with the same speed from his perspective.
This odd, odd although logically consistent fact indicates that time is relative. Time is experienced differently depending on your speed. A measured time of, say, 1 second by the moving observer inside the train would be measure as more than 1 second from the stationary observer. More time went past for the stationary guy. That stationary guy is now slightly older than the moving guy on the train. 
A: I believe what the OP is asking is specifically:

How does it "feel" to be in dilated time?

The answer is it feels absolutely identical - no difference.
Spending 10 minutes near a black hole "feels" absolutely identical to spending 10 minutes in a normal place.
The only difference is that when you return to your friends on Earth, much more time will have passed for them.  (1 year for you, 5 years for them, etc.)
I'm pretty sure that's what OP is asking here.
Say I walk from A to B on a short path.  But you walk from A to B on a long path.  We both walk at exactly 5 mph, normal walking speed.  It "feels" identical to both of us.  And we end up at the same place.  But you walked say 10 miles whereas I only walked 3 miles.
Similarly being near a black hole is a "shortcut through time".  You both arrive back at the "same time" (like, "year 2340") but much more time passed for the others and much less time for you.
Again, I'm pretty sure your actual question here is

How does it "feel" to be in dilated time?

the answer is: identical.  There's quite simply less of it.
The OP understandably thought that all this talk about "time slowing down" might mean that time "seems to run slowly" (like: when you're at the dentist! or in a slow-motion film).  No, it has nothing to do with that.
All it means is when you arrive home, more time will have passed for the other person: that is ALL it means, OP.  Everything else is the same.
A: In General Relativity there is a concept of interval along a world line, which is the same for all observers and is the local time experienced by an observer travelling on that world line. There is no other type of time, although in any particular coordinate system there will normally be three space-like coordinates and one time-like coordinate. The only questions that can meaningfully be asked about time are:
If two observers exchange signals, what will each observe?
If two observers start at the same world point, separate, travel, meet and compare clocks, what will be the results.
Note that neither question involves any concept of time other than local time along a world line.
Locally, the longest worldline between two points is that followed by a free-falling observer. if one observer is in free fall and the other leaves, accelerates to and around the black whole and then rejoins his companion, the clock of the free-falling observer will register more time than the clock of the accelerating observer.
A: Suppose you stand far away from a black hole for 16 minutes, while an astronaut in a rocket goes near the black hole and then comes back. When it is said that "only 8 minutes passed for the astronaut" it means that


*

*the astronaut's mechanical watch will show 8 minutes elapsed, not 16

*the astronaut's electronic watch will show 8 minutes elapsed, not 16

*a grandfather clock on the ship will show 8 minutes elapsed, not 16

*a chemical reaction that would take 16 minutes to finish will come back only half done

*a bacterium on the ship that takes 8 minutes to divide will have divided once, not twice

*if the astronaut measured time by counting heartbeats, they would measure 8 minutes, not 16

*cookies that take 16 minutes, baking in an oven on the ship will be only half done


In short, absolutely everything slows down. So there are only two logical possibilities: either time itself is slowing down, and hence slowing down everything with it, or time always goes at the same rate, but all 7 of the "clocks" above are coincidentally slowed down by the same amount. 
The second possibility is, as you  might guess, much more complicated to work with, so in physics we take the simpler option. We model all 7 of these results by just saying time itself slows down.

It should be emphasized that I don't mean time on the rocket is slower than some "absolute time", i.e. there is no "objective clock" that the astronaut can compare their clock to. Instead I've phrased everything in terms of comparison between the astronaut's clock and your clock, which is appropriate because it only involves actually observable quantities.
Mathematically, in general relativity we model spacetime as a Lorentzian manifold, and that manifold does not come with any structure like an absolute time. There are just points. However, using the metric we can assign an elapsed time along any path, by integrating the proper time. It is this time that I'm talking about being 'faster' or 'slower' here.
