A supernova is the sudden explosion of an entire star. There are at least two general mechanisms that cause a star to explode; these explosions briefly cause the star to emit hundreds of billions of times more light than usual. As such, a single supernova is approximately as bright as an entire galaxy, and the explosion is visible across the entire Universe. It is literally like setting off a flashbulb in the center of a large darkened stadium.

Type Ia is the designation of one particular type of supernova. It is caused by the explosion of a type of star known as a white dwarf. White dwarfs are relatively common in the Universe; they are the end result of the life of a low-mass star like our Sun. Eventually, the Sun will stop producing energy by nuclear fusion when it exhausts its supply of nuclear fuel; it will then expand hugely to become a red giant, to follow with a slow collapse to become a white dwarf -- a very dense star with the mass of the Sun, compressed to the size of the Earth.

Such a star will gradually (over trillions of years) cool down to eventually become a cold, dark sphere called a black dwarf. That is the ultimate fate of our Sun.

Things get more interesting, though, if the white dwarf is one member of a double-star system. Eventually, as part of the aging process, the other star will swell in size, and if the white dwarf is close enough, it can steal mass from the swollen star.

The catch is, however, that there is an upper limit to the mass of a white dwarf. In 1930, Subrahmanyan Chandrasekhar (of Chandra X-ray space telescope fame), worked out that the force that kept a white dwarf from further collapse could only hold against a star of no more than 1.4 times the mass of the Sun. A star more massive than that cannot be a white dwarf because gravity will crush it to overcome the form of internal pressure that holds it out.

So when an accreting white dwarf accumulates enough matter from its companion to exceed this limit, it will begin to collapse, compressing its core to the point of much higher temperatures than those reached in the star during the normal part of its lifetime. This initiates new nuclear reactions which were not previously possible. These reactions proceed explosively as the white dwarf continues to collapse and drive the internal temperature higher. Within just a few seconds, the star explodes.

This is an important event for cosmology because a type Ia explosion always occurs in exactly this way with a star of exactly the same mass, which leads to explosions which are exactly the same brightness (within a certain amount of error due to differences such as exact chemical composition).

Now if you know how bright something is, you can easily calculate how far away it is. So if we observe a type Ia supernova in a distant galaxy, we know how far away that galaxy must be. We can compare that with the distance we calculate from the galaxy's redshift. The difference between the measured and calculated distances can be used to determine how fast the Universe was expanding in the past, since we can figure out what the difference ought to be if the expansion rate is constant, and compare that to the difference we just observed.