What is the fastest algorithm for multiplication of two n-digit numbers?
As of now Fürer's algorithm by Martin Fürer has a time complexity of $n \log(n)2^{Θ(log*(n))}$ which uses Fourier transforms over complex numbers. His algorithm is actually based on Schönhage and Strassen's algorithm which has a time complexity of $Θ(n\log(n)\log(\log(n)))$ 

Other algorithms which are faster than Grade School Multiplication algorithm are Karatsuba multiplication which has a time complexity of $O(n^{\log_{2}3})$ ≈ $O(n^{1.585})$ and Toom 3 algorithm which has a time complexity of $Θ(n^{1.465})$

Note that these are the fast algorithms. Finding fastest algorithm for multiplication is an open problem in Computer Science.

References :


Fürer's algorithm
FFT based multiplication of large numbers
Fast Fourier transform
Toom–Cook multiplication 
Schönhage–Strassen algorithm
Karatsuba algorithm