Why is Earth's Core Iron?
Simply put, the amount of iron in the solar system/universe is higher than nickel due to a process called nucleosynthesis. 

Basically at it's origin the universe comprised only of hydrogen, through nuclear fusion heavier elements were/are being created. Fundamentally, hydrogen is fused into heavier elements in the star, our sun is currently turning massive amounts of hydrogen into helium through the proton-proton chain.

Eventually as a star gets older you start seeing fusion of heavier elements, helium, carbon, nitrogen, etc up until you hit iron at which point you hit what is known as the iron peak  where fusing iron atoms together requires more energy than is release. Heavier elements are all formed during supernovas and thus we see lower level of abundance. That being said the abundance of nickel is still higher than all the other heavier metals.  

https://en.wikipedia.org/wiki/Stellar_nucleosynthesis

http://physics.info/nucleosynthesis/