What is Olbers' paradox?

Olbers' paradox is a thought experiment that arises in cosmology* which shows that our universe does not fit the "Static Newtonian" model.
Sir Isaac Newton came up with a model for the universe which can be described by four basic statements:
i) The universe is infinitely large
ii) The universe is infinitely old
iii) The universe is not expanding or contracting (i.e. it is "static")
iv) All the planets, stars, galaxies etc. are roughly evenly distributed throughout the universe (i.e. it has "large-scale homogeneity")
This came to be referred to as the Static Newtonian model.
Olbers' paradox is a logical argument which attempts to show that our universe cannot fit this model. It does this by arguing the following:
If the universe was infinitely large, and had a fairly even distribution of stars throughout it, then this would mean that there would be an infinite number of stars surrounding the Earth on all sides. Furthermore, if this universe was infinitely old, then the light from all of those stars would have had enough time to reach the Earth. With an infinite amount of stars on all sides, constantly emitting an infinite amount of light, our sky would always be infinitely bright - there would be no difference between day and night. This is obviously not the case!
We can see that our four statements have lead us to a model that does not agree with our observation that the sky is dark at night. Therefore at least one, or possibly several of these statements must be false. This leads us to the following conclusions: either,
a) the universe is finite in size, and thus has a finite number of stars
b) the universe has some finite age, and thus the light from most stars has not had enough time to reach us
c) the stars, planets etc. are not evenly spread on a large scale, which would also allow for a finite number of stars
d) some combination of a), b) and c).
These conclusions lead us to consider more sophisticated models of the universe, such as the Big Bang model, which states that a) and b) are true, but not c): i.e. the universe is expanding (and therefore not static, unlike Newton's model).
 
 
*The study of our universe; its structure, how it changes with time, how it began, etc.