Space is expanding.
The universe? we don't know -- all we can observe, and measure, is that SPACE is expanding.
Everywhere. Even inside individual photons of light. Inside atoms, inside you and me, between stars, between galaxies...
It is not that things are moving away from each other. It is rather that more space is being added everywhere, all the time.
Locally, the rate is extremely small. For example, during a 70-year lifetime, the amount of space being added over a length of 2 metres (a somewhat tall man), adds up to almost the size of one atom.
Space is expanding everywhere. Over long distances, it does add up. Over a distance of one MILLION parsecs (3.26 million light-years), the amount of space being added comes to approximately 70 km every second.
This expansion is the same everywhere, therefore the amount being added is "linear" -- translation = if you double the distance, you get twice as much "new space" every second.
Two million parsecs = expansion of 140 km/s
This sounds like a lot, but if you could imagine the size of a million parsec (more than the distance from here to the Andromeda galaxy), you'd see that the rate of expansion is extremely small.
And yet. If you apply this rate backwards (going back in time) to two points separated by a million parsecs, you'd see that the distance between them reaches zero, when you apply a time duration of 13.8 billion years (the real calculation is done in seconds, but you get the drift).
If you take two points further apart? Then you need to account that the rate is linear (since the expansion is cumulative over distance). Two points separated by two million parsecs would see 140 km of new space being added every second. Calculate backwards until the distance reaches zero and... you get the same 13.8 billion years.
--What about the non-visible part of the universe?
Of course, things COULD be different beyond our horizon (outside the Observable Universe) but, for now, we have no reason to think so. It is like waking up on a ship in the middle of the ocean. You can see water all around and, climb as you may, you can only see as far as the horizon. If the highest place you can climb is 400 feet above the water line, then your horizon would be at roughly 22 nautical miles (approx. 43 km, the length of a marathon race).
If you are in the middle of the Pacific Ocean, this is nothing. However, by studying the ocean within this 43 km radius, there are things you understand. For example, the link between the speed of the waves and the surface tension of the water; they determine how energy is transported on the surface of the ocean. Strong binoculars may not allow you to see the surface beyond the horizon (just like telescopes do not let us see outside the Observable Universe), but they show you that the waves coming over the horizon seem to behave the same way than the waves close to your ship. And this is true regardless of the direction.
You conclude that the ocean, outside your 43-km radius, probably looks a lot like your little patch of water.
We do not know about the whole universe. However, based on observations made with the WMAP a few decades ago, it looks like the "outside" universe behaves a lot like our little patch of Observable Universe, at least for a distance three times our horizon radius. If that is so, then we know that 13.8 billion years ago, all distances between any two points within that region would have been zero. Whatever our part of the universe really is, it could not have existed in any state that we would recognize before that moment 13.8 billion years ago.
WMAP = Wilkinson Microwave Anisotropy Probe