The expansion of the Universe accelerates
Particularly in one respect, this property of Type Ia supernovae has been ground-breaking in astronomy, more specifically in cosmology.
The reason is that they have enabled us to determine the distances of galaxies far, far away, where we at the same time could measure their velocity with respect to us through their cosmological redshift.
And why is that interesting?
Well, we know that the Universe expands. We've known that since Edwin Hubble and others in the 1920's measured distances and velocities to galaxies in our "neighborhood".
With Type Ia supernovae we can investigate the relation thousands of times farther away than Hubble did.
Because it has taken the light some times to reach us, that in turn means that we can see how fast the Universe has expanded through most of its history, billions of years back in time.
And here, astronomers discovered something
terribl… something very interesting in the late 90's:
In the beginning of its life the Universe expanded incredibly fast, but the gravity from everything in the Universe (primarily your mom) gradually decelerated the expansion.
Whether the gravity would one day decelerate enough to make the Universe contract, or whether it had received a large enough kick in the beginning to expand forever, depends on how much stuff there is in the Universe.
Both ordinary matter (such as stars, planets, and bicycles), dark matter, and in fact also radiation, helps slowing down the expansion.
What the observations showed us was that the Universe indeed did slow down in the beginning.
But relatively recently (cosmologically speaking — it's roughly 3½ billion years ago), something peculiar happened: the Universe started speeding up.
And not only did it speed up, its expansion accelerates, so it goes faster and faster.
We don't know with certainty the reason for this, but the most "mainstream" explanation is that it's a property of empty space itself.
Various possible mechanisms go by the name dark energy.
The more the Universe expands, the more empty space is created, and hence the more dark energy, which makes the Universe expand even faster.
In other words, the Universe seems to keep on expanding forever, so everything gets farther and farther away from each other
(it might reassure you that this is only on "cosmological scales"; galaxies, stars, and planets are held together by gravity, and you are held together by electromagnetic forces.