The “ordinary Big Bang and expansion” (no inflation, no dark energy)
Einstein’s General Relativity allows a solution (FLRW metric) where an empty Universe expands. Imagine that you have particles (say, hydrogen atoms) in a cubic grid with exactly 1 light year between nearest particles. (We assume that their mass is so tiny that this Universe essentially behaves as if it is empty). And after each second distance between each particle increases by 1 meter. Not because they move, but because space “grows”. That’s that solution.
Even though empty Universe is expanding without slowing down (distance between test particles grows by exactly one meter per second, all the time), note that if you look back into the past of this Universe there were times when test particles were much closer together – say, only a billion km apart – and they moved away at exactly one meter per second too. It’s clear that there was a moment where they had to be zero meters apart. That’s the “Big Bang” moment. The moment itself is problematic (singularity!), but any moment after that is not. For example, one microsecond after this moment test particles were 1 micrometer apart. It’s a bit curious that in this scenario expansion seems to be very rapid at first (“density” of test particles falls very quickly), and millions of years later, it looks very gradual, but expansion rate is in fact constant.
Big Bang with accelerating expansion
Now, if you use General Relativity equations with cosmological constant Λ > 0, the picture changes. Grid of test particles grows not by exactly one meter per second all the time. Now it grows faster with time. If Λ is very small, at first speedup is not noticeable, but later it will be: test particles will not only move away from each other, they will seemingly do it faster with time.
That’s one possibility what dark energy is – maybe it’s just Λ. However, GR with Λ = 0 but with some other field permeating all space and having appropriate property (negative pressure) will have exactly the same behavior.
Big Bang with matter
How presence of matter changes this? FLRW metric with homogeneous distribution of matter will expand too, but expansion will slow down. (Heuristically, “matter will attract itself and try to shrink the Universe”). If there are lots of matter, expansion rate can even go to zero and expansion starts to go backwards. Between “too little matter, eternal expansion with nonzero rate” and “too much matter, expansion stops and reverses” there is a borderline case where expansion never stops, but its rate falls ever lower, tending to zero with time, but never reaching it (that’s “critical density Universe”). This all was about “normal” matter, with positive pressure. With “negative pressure matter” the effect is opposite – this was already described in the previous paragraph – that’s “dark energy”.
Big Bang with inflation
And finally, what if dark energy field is variable (e.g. it has several possible stable values) and one of these values is large (or there may be several such fields)? Alternatively, what if Λ can not only be larger than zero, but can be VERY MUCH larger than zero? Nothing unusual will happen, the Universe will behave as described above: grid of test particles grows faster with time. But very, very much faster. That’s inflation. (If you have a separate “large value dark energy field” for it, that’s “inflaton field”).
If you have inflationary Universe, even with matter, it expands astoundingly quickly, essentially becoming empty within a very small fraction of one second. And if then suddenly Λ (or dark energy) goes down to a very small value, you get an empty symmetrical flat expanding Universe. If “Λ going down” releases energy (in a form of appearance of new particles everywhere), you get a NON-empty symmetrical flat expanding Universe. As of 2016, this model fits observational data rather well.