What do graphene "armchair(5,5)" and "zigzag(5,0)" refer to? I am reading some articles about the electronic properties of graphene and while reading it I came across graphene armchair and zigzag structures, which have different bond structures. 

The two different orientations (armchair and zigzag) are due to the arrangement of carbon atoms in the periphery (wiki), but I am unable to understand what armchair(5,5) and zigzag(5,0) mean.
The third picture, subpic (d) here shows you the geometric armchair(red) and zigzag(blue) structure:



It's derived from  structure description of Carbon nanotubes similar to surface reconstruction/overlayer structures:


  The integers n and m denote the number of unit vectors along two directions in the honeycomb crystal lattice of graphene. If m = 0, the nanotubes are called zigzag nanotubes, and if n = m, the nanotubes are called armchair nanotubes. Otherwise, they are called chiral. The diameter of an ideal nanotube can be calculated from its (n,m) indices


I hope you have some background, otherwise here is a nice visualisation, or look the lecture linked above. It's important to understand what the unit cell of a 3D crystal or 2D surface is.