Is a black hole a 3D hole? And doesn't it pull into the 4th dimension? Hear me out here please:
A body on a line (a 1-D world) causes a warp in the line, i.e. a curve (2-D)
A body on a plane (a 2-D world) "sinks", causing a warp in the plane, i.e. a pit (3-D)
Then does that mean that a body in a 3-D world warps space into 4-D?
 A: To expand a bit on the others' answers, a couple of ways to visualize/understand the difference between intrinsic and extrinsic curvature:
Intrinsic curvature, as the name suggests, only deals with stuff that lies inside a surface/space/manifold etc. (I will use the term manifold)
If your lines, triangles, etc. don't work the same way as they do in euclidean geometry, then you are dealing with intrinsic curvature.
An example of intrinsic curvature: Take a spherical object, and grab a pen. Designate a pole on the sphere, and the associated equator. Take an arbitrary point on the equator, and draw a line along the great circle connecting that point with the pole, then turn 90°, and continue the line you just drawn along the great circle designated by your new direction, until you hit the equator again. Now connect the two points on the equator by a line along the equator!
What you got is a "triangle" on the sphere, whose sides are great circles of the sphere, and the total sum of the internal angles are 270°. This is decidedly not euclidean, and this means the sphere has intrinsic curvature.
Of course, the sphere also has extrinsic curvature, but note, that a "flat mathematician" living on the spherical surface, will not be able to see how the sphere bends in 3D, since he would not even perceive 3D. However, he still would be able to measure angles, so he would still arrive at the conclusion, that this triangle has an internal angle sum of 270°, and thus he lives in curved space.
(I will note, without proof, that the geodetics, aka. shortest possible curves connecting two points, on a sphere are always the great circles, meaning that the above described triangle actually has "straight" sides. The flat mathematician would perceive them as straight.)
Extrinsic curvature, on the other hand measures how a manifold bends in a higher dimensional manifold it is embedded in. A "flat mathematician" has no chance of ever measuring the extrinsic curvature of his space, and because we cannot see outside our spacetime, it is not really meaningful to consider whether our spacetime is embedded in a higher dimensional spacetime or not, since even if it was, we could not measure it.
An example of a surface with extrinsic curvature, but not intrinsic: If you take a sheet of paper, and roll it up into a cylinder, then obviously your cylinder has curvature, but if you perform experiments, like what I described with the sphere, you will see, that everything on the cylinder works exactly as you'd expect it.
For example, if you draw something on the plane, and you roll it up into a cylinder, you1d drawing would not be distorted.
On the other hand, if you draw something on a plane, and crumpled the plane into a sphere (also note, that you cannot do that without tearing or deforming the paper! This is exactly why maps of the Earth look funny, when the map is flat, and not an earthglobe.), then your drawing would end up with distorted distances and angles.
Some math (but only in words):
In general relativity, the curvature of spacetime is intrinsic curvature. You cannot associate any usual down-to-earth notions of curvature to it. This curvature is represented mathematically by a tensor field called Riemann curvature tensor, in physicists' notation, generally looking like $R^a_{\ \ bcd}$.
I will not give an explicit form of the Riemann curvature tensor, but I will say, that you can construct the curvature tensor in multiple ways, one of them being a measure of the deviation of geodetics (the afore mentioned straightest possible curves), meaning that it measures how initially parallel geodetics drift apart (or converge to one another) as you go along them. It is not hard to convince yourself, that this is indeed a good way to measure intrinsic curvature, since in Euclidean space, initially parallel straight lines will always be parallel.
It does not mean that extrinsic curvature has no place in general relativity, but it is present only in some more advanced formalisms, and even there it is not the extrinsic curvature of spacetime, but the extrinsic curvature of hypersurfaces embedded within spacetime.
Mathematically speaking, the extrinsic curvature measures, that if you have a vector field that is tangent to the hypersurface everywhere, and you differentiate this vectorfield also in a tangential direction, then how much of the resulting vectorfield is normal to the surface. Obviously, this requires an ambient space to exist, otherwise there is no sense of "normalness".
I hope this helped.
A: A black hole is a 4D object, but that's because all objects are 4D as they live in a four dimensional spacetime - three spatial dimensions and one time dimension.
However what I suspect you're asking is whether there has to be an extra spatial dimension for space to bend in, making five dimensions in all. If so, then the answer is that no there is no extra dimension.
The pictures you've seen of the rubber sheet models for spacetime model the curvature as extrinsic. There is a precise definition of extrinsic curvature, but for our purposes extrinsic curvature is curvature into another dimension outside whatever you're curving. So the 2D rubber sheet gets deformed into a third dimension.
However in general relativity the curvature is intrinsic. This is hard to visualise, but suppose you left your rubber sheet flat, but stretched it in a direction that lies in the plane of the sheet. This is intrinsic curvature. I discuss this in my answer to Universe being flat and why we can't see or access the space "behind" our universe plane?, and there is a related discussion in What is the universe 'expanding' into?.
A black hole forms by intrinsic curvature of four dimensional spacetime (remember that it curves time as well as space, which if you've seen Interstellar is what caused the time dilation for the astronauts near the black hole). So there is no extra, fifth, dimension associated with it.
A: Is a black hole a 3D hole?
I think so. Some people will say it isn't really a hole, but I think it is. That's because I think the "frozen star" interpretation is the one that's right. You can see a mention of that in Kevin Brown's article The Formation and Growth of Black Holes. He doesn't favour it, many people don't know about it, and others maybe hate it, but I'm hoping that will change. Anyway, I also think it's something like the gravastar which features a "void in the fabric of space and time". Have a look at the gif on the Wikipedia black hole article. The original image was by Paris cosmologist Alain Riazuelo. It looks like a bullet hole in a black car.  
And doesn't it pull into the 4th dimension?
No. It just pulls things towards it, in the ordinary 3D spatial dimensions. Because it has a gravitational field. What's interesting though is that analogies like the rubber sheet are back to front. See the stress-energy-momentum tensor? Note the energy-pressure diagonal? That's pressure, not tension. To improve the rubber-sheet analogy you replace the sheet with a solid, a bulk. And then instead of pulling it in like the picture on the right here, you push it out. Then to avoid getting confused by the curvature of the Earth/star/etc, you zoom in, like this: 
 
What you can then see is what's called Ricci curvature, wherein a volume deviates from the Euclidean-space norm. Note that space itself isn't curved. Spacetime is curved, but space isn't, see Baez re that: "Similarly, in general relativity gravity is not really a `force', but just a manifestation of the curvature of spacetime. Note: not the curvature of space, but of spacetime. The distinction is crucial.". 
Then does that mean that a body in a 3-D world warps space into 4-D? 
No. See Einstein's Leyden Address. A concentration of energy in the guise of a massive body "conditions" the surrounding space, altering its metrical properties, such that a gravitational field is a place where space is "neither homogeneous nor isotropic". We then "describe its state" using ten functions that are commonly know as the metric tensor or just curved spacetime. This combines space and time, but space isn't warped or curved. Your plot of spatial properties, made with say light clocks in an equatorial slice of space through the Earth, is curved instead.     
