Let's use a space-time diagram.  Time runs vertically, space runs horizontally.  On this diagram I've drawn two points, each representing an event.  An event means "something happens at this point in space at this point in time", which is a unique location in space-time.  On this diagram, Event 2 happens in the future from Event 1, and in space further to the right than Event 1.  

I also put in yellow lines leading from event 1 to represent the speed of light (imagine event 1 is a flash grenade going off).  In a space-time diagram we often choose units of space and time so that the speed of light is a 45° angle (for example, 1 light year of distance in 1 year of time), so this pulse of light extends upward as a 45° cone.  A.K.A a "light cone".  Since nothing can go faster than light, all allowed paths from event 1 must lie inside the light cone.



Finally, I've drawn two possible paths that a person could follow to get from the first event to the second event.  Both paths are at speeds slower than light, but one is more direct.  Path B in blue moves at constant velocity, while path A starts off moving to the left, then slows down and changes direction to reach Event 2.

Question:  Which path is shorter?  (Has the shortest distance in space-time?)

The counter-intuitive answer:  Path A is shorter!  

In Euclidean geometry, a straight line between two points is always the shortest.  But space-time obeys a different geometry.  Instead of the Pythagorean theorem where distance is defined as sqrt(x^2 + y^2), distances in space-time are sqrt(t^2 - x^2).  That minus sign makes all the difference.  Instead of the most direct path between events having the shortest distance, it's the longest distance!  Any change from a straight line between two events will make the length of that path shorter!  And if your change in x equals your change in t (such as moving 1 light year in space in 1 year of time, meaning moving at the speed of light), then the length of the path is zero.  (That's right, those two yellow lines have zero length!)

And the final bit of insight:  What we perceive and measure as time is not the time axis.  Clocks do not measure t!  What they measure is the distance they have traveled in space-time, t² - x².  That means if you move from Event 1 to Event 2 on the direct path B, you measure the longest time elapsed on your clock!  If you instead follow the less direct path A, you will measure a shorter time!  Or if you move at the speed of light between two events (you can't actually, but we can consider the limit that your speed approaches c), then your clock will never tick!  The entire trip goes by in an instant according to you.

It seems like more you're inside a curvature of space more you move faster inside time. Actually, you also move faster inside space, so it makes sense...?

It's the other way around!  Clocks tick more slowly in a stronger gravitational field (more space-time curvature).  Time at sea level on Earth passes more slowly than time on top of a mountain.  But you're also not moving in space.  If we're using coordinates fixed to the Earth, then as you sit on the surface you are not changing your position.  Your change in x is zero.  You are moving in time, but again what you measure on your clock is the distance traveled in space-time, and because that space-time is now curved that distance is shorter than it would be in flat space-time.  Or put another way, that slower passage of time on a massive object is a direct manifestation of the curving of time by the gravitational field near that mass.

Well, if you was in complete void inside intergalactic space and no gravity source from any direction you should stay still in time too, right?

Again just the opposite.  If you're floating still far from any gravitational sources (flat space-time), then you are not moving in space and only moving in time.  Since the time you measure is sqrt(t² - x²), and you're only moving in t, your clock ticks at the fastest possible rate (compared to anything else in the same flat space-time).  Any movement you make will change your x and make your clock tick slower.

Put another way, clocks always tick the fastest in a frame of reference where the clock is at rest (i.e. you move with the clock).

Probably not, instead the illusion is that parts of the universe that contains no visible matter are the ones that expands faster, and they expand faster and faster more is the void that they contain. Probably because they mainly contain dark energy so there is not an opposite force from baryonic matter's curvature of space that opposes the "negative pressure" of dark energy.

The whole universe expands uniformly at large scales.  Pick any two points sufficiently separated in space, and the rate at which they recede from each other is proportional to the distance between them.  Locally, matter in the form of galaxies and clusters of galaxies "break off" from the expansion, because they are bound together by gravity.  So galaxy clusters and everything within them do not expand at all, while the space around them expands and drives the clusters apart.

The expansion itself is a consequence of the Big Bang.  If the universe was empty then that expansion rate would stay at the same value forever.  Introducing matter and radiation pressure to the universe acts to slow the expansion rate down, while dark energy causes it to speed up.  

Consider a box filled with some molecules of air.  The air is well mixed, and if you can watch it on the molecular level you see them randomly bouncing around and colliding off the walls and each other.  But if you record a movie of this and then play it backward, it basically looks the same.  Or if someone gave you a movie of molecules bouncing around in a box, you would not be able to tell if it was played backward or forward.  The direction of time in this system is ambiguous.

Now let's suppose we divide the box in two with a barrier, and put all of the air on one side.  What happens if we then poke a hole in the barrier?

Common intuition says of course the air will flow through the opening until the pressure equals out, with equal number of air molecules on each side.  Ah, now we have a direction of time!  If you see a film where the air is uniformly spread through the box but then suddenly all moves to one side of the barrier, you know that the film was played backwards!  

Why?  What law makes it so that time flows one way and not the other?  The answer is the 2nd law of thermodynamics, which arises from the statistics of how many different possible ways there are of arranging a system.  Let's make this simple, using the box example:

Imagine we have just 2 air molecules.  Or just 2 particles.  How many ways can they be arranged in this split-box?  Call one of the particles "A" and the other "B" 

1)  AB both on right side of barrier.
2)  A on right B on left
3)  B on right, A on left.
4)  AB both on left.

So if we have 2 particles to distribute in the box, there are 4 distinct possibilities.  Of those 4, two of them have both on the same side.  So if you start with 2 particles on one side, open the barrier, and then check again a long time later, you have a 50-50 chance of seeing both particles on the same side vs. one on each side.  

Now let's make it 4 particles:  A, B, C, and D.  Here are their possible distributions:

| ABCD
D | ABC
C | ABD
B | ACD
A | BCD
CD | AB
BD | AC
BC | AD
AD | BC
AC | BD
AB | CD
BCD | A
ACD | B
ABD | C
ABC | D
ABCD |

2^4 = 16 possibilities:
All particles on one side:  2/16 = 13% probability
3 on one side: 8/16 = 50% probability
Evenly spread: 6/16 = 38% probability.

So we see that having more particles makes it less probable to find them all on one side.  The unambiguous arrow of time appears!  Opening the barrier and checking sometime later we naturally expect to see the particles more evenly spread.

Let's make the number of particles even greater, so the power of large-number statistics really takes hold.  To make a computational shortcut, for a number N particles, the number of ways that K of them can be found on one side is found by "N choose K".

Let's make it 10 particles.  Then there are 2^10 = 1024 ways of distributing them in the box.  What are they?


All on one side --> 2*(10 choose 10) = 2 ways          0.2%
9 on one side --> 2*(10 choose 9) = 20 ways            1.9%
8 " " --> 90 ways                                                     8.8%
7 --> 240 ways                                                       23.4%
6 --> 420 ways                                                       41.0%
5 (Evenly split) --> 252 ways                                   24.6%

Now we see it is much less probable to find all of the particles on one side.  The distribution trends towards having them closer to evenly split.

This is the origin of the arrow of time in a (very simplified) nutshell.  Systems tend to evolve towards states where there is a greater number of non-distinguishable configurations of the microstates -- that is, for the entropy to increase.  It's also the reason heat flows from hot objects to cold objects rather than the other way around.  Any process that increases entropy, or keeps it the same, is preferred over a process that would decrease it.

Roger Penrose is famously interested with these deeper principles behind the arrow of time, as are many others who seek to understand why the Big Bang happened as such a low entropy condition.