By observing type Ia supernovae, it was discovered that the expansion of the universe is accelerating [cite]. The diagram on the top left is called a Hubble diagram and it plots the magnitude (which tells us the distance) vs the redshift (which tells us how fast it is moving away). The bottom image is a plot of residuals. The y-axis is the difference between observed magnitudes and the expected magnitudes for 3 different cosmological universes. The three lines represent three different compositions of the universe of matter and dark energy.
An object's distance can be determined by measuring the observed brightness and comparing it to its intrinsic brightness. In order to do this, the intrinsic brightness must be known, which is not easy for any star or object. This is because many other factors that are not easily measurable on their own can influence intrinsic brightness of objects.
Intrinsic brightness of a type Ia supernova can be determined by measuring how it evolves through time. The shape of a Ia supernova light curve, or the speed at which it fades away, is related to its intrinsic brightness.
The plot on the top right shows the many different light curves for many observed type Ia supernovae with different durations and brightness. Most of the observed supernovae fall on the middle yellow line, and the others can be normalized. The bottom plot is all of the supernovae light curves normalized to fit on the same line.
Perlmutter, 2003