How is the runway pavement chosen?
I am neither builder nor airport manager and can't provide details, but as there is no answer yet, I'll try to summarize the main differences between the materials:


Concrete can support highest loads and lasts long. It is however most expensive to build and when it eventually gets damaged, it can't be patched and has to be rebuilt in large slabs. It also requires stable base, because it is rigid. It is built in slabs with spaces between them that are required due to account for thermal expansions and give somewhat bouncy ride.
Asphalt. There is a big variation between different kinds of asphalt concrete, but it is generally cheaper than concrete, but can't support as big loads and is susceptible to damage by frost. Asphalt is never completely rigid and slowly deforms under load. This allows using it on unstable base, but it means bumps form by prolonged use reducing the total lifetime. It also easily cracks and when water freezes in the cracks, potholes quickly form. They can however be relatively easily patched. It does not require any thermal expansion gaps and provides smooth ride.
Asphalt and concrete can be combined. A concrete with a layer of asphalt on top provides the strength of concrete and smooth ride of asphalt. The asphalt also protects the concrete base from freezing water while itself can be easily patched. This is lately common on roads (where concrete alone is really uncomfortable), not sure how much on runways.
Gravel is simple to build; all you need is some stone from nearby quarry and a steamroller. The main disadvantages are risk of foreign object damage by loose stones thrown up by the wheels and need of regular maintenance. So it is used mainly in remote cold places where concrete would be too expensive and asphalt would not last long in prevailing weather conditions.
Rolled earth with or without grass is even simpler then gravel, used for small airports where big loads are not needed.
When runway is listed as grass, it means it is not even rolled. Usually enough for small airplanes and obviously cheapest.
Other surfaces usually as provided by local conditions (snow, salt etc.)


For every particular airport, the investor evaluates the available technologies whether they can handle the expected traffic (to handle large jets like 747 or A380 concrete or concrete-asphalt combination is preferred), whether they are buildable in given location (the ground has to support the weight too) and how much building them will cost and choose the option that satisfies the requirements and they have money for.