How can time be curved? Time isn't a physical object, but according to Einstein's theory of gravity, mass bends spacetime towards things with mass and makes them fall. How does a physical object affect something intangible?
 A: In a general sense, time is just a dimension. We could draw a diagram where we say an object curves in space over time. But we could also flip the diagram and say the object curves in time over space. It's just a matter of perspective.

I think a better way to think of the "curvature" of spacetime is to think of its density changing. Spatial density pushes objects from high density to low density (gravity). Time density tells us things tick at a different rate relative to the "universal clock". Lower temporal density means fewer ticks update an object in the same span of "real" time than an area with higher density.
Of course, "real time" is a hard thing to pin down, and some would argue it's meaningless. But I've always found it easier to acknowledge that there's some relativity to reference frames than to try completely removing them from conversation.
A: To be precise, time is not curved, it is the spacetime manifold that has curvature. Such a manifold can be given a coordinate chart - an arbitrary one - that assignes to different points (spacetime events) some labels.
It turns out that we need one time coordinate to label the time slices and three spatial coordinates to distinguish the events within that slice.
What you might satisfactorily say is that the rate of flow of the proper time - the time that your own clock shows - depends on how much spacetime curvature is in your vicinity. This is a quantity independent of any coordinates, a true physical property if you will. This proper time is a measure of your aging, biological processes in your organism, the oscillatory frequencies in atomic transition in the atomic clock you might have etc.
Imagine you and your friend orbiting a static black hole. You get the same clocks and synchronize them. They show the same time and tick at the same rate.
Then your friend stays at his orbit (at a distance $r_{friend}$) and you go much closer to the black hole (at a distance $r_{you}$).
You spend some time there and upon returning, you make a discovery that your clocks show different readings!
In fact, using the Schwarzschild solution you could predict that your clocks will exhibit a ratio of proper times that have passed equal to
$$ \frac{\tau_{you}}{\tau_{friend}} = \frac{1-2M/r_{you}}{1-2M/r_{friend}}$$
In the above, I omit the problem of determining when you start and stop the readings and how you account for the travel back and forth. It does not change the fact that your clock will exhibit fewer ticks-and-tocks when you compare it with the one of your friend.
A: If you think about it,  "space" is no more tangible a concept than "time" is.  If one can curve, so can the other.
Spacetime is curved in the same sense that the surface of the Earth is curved.   Two ships sailing on initially parallel paths may nevertheless converge and meet.
An similar example of time curvature is, if someone on a high altitude point, like the Red Bull guy, flashed a laser pulse down at the ground, and then flashed one again 1 second later (by his clock), the pulses would arrive on a detector on the ground less than 1 second apart by our clock. Less by roughly $10^{-12} \,\rm{sec}$.  So the time interval between the two pulse events decreases as we approach the center of the Earth, just as the space interval between the ships decreases as they approach the North Pole.
If it were a black hole instead of the Earth, the pulses could go a lot deeper into the gravity well, and the time between them would approach zero.
A: 
How does a physical object affect something intangible?

This is actually quite common, not only for the gravitational field,
but also for other fields.
Take for example the electromagnetic field.
This field carries not only energy (as everyone knows from sunlight),
but also momentum and angular momentum.
Hence you should think of the field not as intangible,
but more as a real physical object.
Electric charges make an electromagnetic field.
And the electromagnetic field affects the movement of charged bodies.

Time isn't a physical object,

More precisely you should talk about spacetime instead of time,
because space and time are closely intertwined.
The same way of thinking like above for the electromagnetic field
applies for the gravitational field
(or for spacetime curvature, by Einstein's language).
Thus spacetime should be considered a real physical object.
Masses curve spacetime.
And curved spacetime affects the movement of masses.
In a sense you can think of spacetime as a very hard stuff,
because you need enormous masses to make only
a tiny curvature of spacetime.
For example: at the surface of the earth the gravity
acceleration is $g=9.8$ m/s$^2$. According to Einstein
this corresponds to a tiny curvature of spacetime.
Its curvature radius is actually $r=\frac{c^2}{g} = 9.2\cdot 10^{15}$ m.
