"The Universe is all there is, or was, or will be." That seems like a reasonable statement to make, doesn't it? It certainly matches with our conception of the word Universe, which implies that this is all of space and all the matter and energy within it. We certainly live within the Universe, and can see an enormous amount of it: some 46 billion light years in all directions. After 13.8 billion years since the hot Big Bang, and the fabric of space expanding for all that time, this is the absolute limit of how far away we can see.
But what lies beyond that? Is there more Universe like our own? The answer is yes, there ought to be. But there ought to be something even more than that: a larger spacetime structure that has an enormous, countlessly large number of Universes embedded within it. If our best theories are correct, our one Universe is not enough. Here's why.
Imagine you went all the way back to the start of the Universe as-we-know-it: the beginning of the hot Big Bang. What would it look like? You'd find yourself immersed in a hot bath of particles, antiparticles, and radiation. They would all be massless, and therefore moving around at the speed of light, since the Higgs hasn't yet given mass to the Universe. And the temperature and energy of these particles would be exquisitely high: around 1028 K, give or take a little bit. Everything that was energetically allowed would exist, and particle collisions — including spontaneous creation and annihilation of particle/antiparticle pairs — would happen rapidly, frequently, and relentlessly.
From this point on, we normally run the clock forward in time, watching the Universe expand, cool, create more matter than antimatter, and eventually form nuclei, atoms, gravitational clumps, stars, galaxies, and after enough time, human beings.
But what if we went the other way? What if we went backwards, instead, and asked where the conditions we call the hot Big Bang originated from? We'd arrive at a strange, almost alien-like state of affairs: a period of cosmic inflation. Instead of the energy in the Universe being bound up in particles, antiparticles, and radiation, it would instead be energy inherent to the fabric of space itself. The consequences of cosmic inflation are profound, but not always intuitive.
If the Universe has all of its energy wrapped up in a form that's inherent to space itself, it still expands, but it doesn't cool or become less dense like the post-Big Bang Universe does. Instead, it expands exponentially, which means that it doubles in size after a small-but-fixed amount of time goes by: somewhere around 10-35 seconds. Therefore, by time 10-34 seconds have gone by, the Universe has increased in size by a factor of 210, or 1024. By time 10-33 seconds go by, it's increased by 2100, or ~1030. And so on.
The key realization of inflation is that the energy density remains constant. As the volume of the Universe increases, the energy inherent to space in every region remains the same. As the Universe inflates, it simply creates more and more Universe that's still inflating.
"So what," you say, "it's still all just the one Universe, right?"
Maybe it is. The volume of the Universe increases tremendously, relentlessly, and without bound, but not indefinitely. It only does this until inflation comes to an end. And when it does end, all of that energy inherent to space gets converted into particles, antiparticles, and radiation: the end of inflation coincides with the start of the hot Big Bang.