Why are aircraft turbine blades made of nickel alloys?
Not all turbine blades are made of nickel alloy, some blades are made of such family of alloys but clearly not all. Some blades are actually made of ceramic materials.

The turbine is always located after combustion chamber and the temperature which the first stage of turbine blades is able to resist is a proof of high technology engine. Those blades are exposed over to 2000ºC; the melting point of steel is around 1100ºC, and nickel can go much further up to 1700ºC. Ceramic can go even further. Note that there is also a significant difference between external flow temperature and surface temperature.

Using a high melting point material is one way to solve the problem, there are also ways to reduce the effective temperature over the surface of the metal like using film cooling.



In the figure I introduced above you can see holes in the blade, where "cooling air" (maybe at 400ºC) flows out of the blade creating effectively a film that goes around the blade protecting (cooling) it from external temperature.

However a metal exposed at a high temperature (below the melting point) is not behaving in the same way as at normal temperature. There are some phenomena affecting the materials like creep. When a material is exposed during long term to high temperatures close to the melting point and high stress at the same time (this is important) it starts suffering deformations that lead to a failure of the metal.

So nickel alloys, apart for having a higher melting point are also having one of the best behaviours in creep environments.

The last stages of the turbine rotors, and more easily stators (less demanded in stress) are less exposed to creep conditions and are not needed to be manufactured in a nickel base alloy. 

Just to add as well, in order to reduce creep another approach followed (combined with previous) is to solidify the metal ensuring having a single crystal metal (instead of being a polycrystalline metal). This is utmost art of material engineering (from my point of view).