Observations of Type 1a supernovae in 1998 and 1999 gave us our first evidence that the universe is expanding at an increasing rate. Type 1a supernovae are standard candles in cosmology. They give off a certain known and consistent amount of light because their mechanism is always the same. Because they are always the same brightness they can be used to accurately measure their distance from Earth by their relative dimness, and they are very bright so they can be seen from great distances. The following 11-minute video by PhysicistMicheal demonstrates how this process works in detail.
These supernovae ultimately give us direct evidence that the universe is expanding at an accelerating rate. Again, this has to do with redshift but there are subtleties about red shift we must understand first.
This kind of data comes from examining the redshift of the photons coming from these supernovae. To see what redshift looks like, compare the black absorption lines in the optical spectrum of a distant supercluster of galaxies (right) to the Sun's spectrum (left). The arrows indicate how much the spectrum is redshifted.
This evidence required far more precise observations and from objects much further away in space in order to compare expansion rates over time. It came very recently (1998) from two teams of researchers making up the Supernova Cosmology Project and the High-z Supernova Search Team. They carried out extensive surveys of Type 1a supernovae, going from those closest to us to those extending up to 9 billion years in the past. Because light travels at a specific velocity in a vacuum, you can measure distance as time - this is how you get a light-year - the distance light travels in one year, so these teams mapped supernovae as distant as 9 billion light-years away from us. This way they were able to link Hubble's law to an accelerating expansion of the universe. They were awarded the Nobel Physics Prize in 2011 for this breakthrough work. To really get a good idea of how they designed and carried out their work, check out this website created by Brian Schmidt, the group leader for the High-z Supernova Search Team. He goes through all the science behind this breakthrough in an easy to understand format made especially for non-scientists, and with lots or helpful diagrams and animations.
The second line of evidence comes from the WMAP (Wilkinson Microwave Anisotropy Probe) data itself, next in Dark Energy Part 9.