What does $\mathrm{A}_r(\mathrm{H})$ represent when calculating the average natural weight of an elemental atom?
The relative atomic mass $A_r$ is 'the ratio of the average mass of atoms of an element ... to 1/12 of the mass of an atom of carbon-12' (from http://en.wikipedia.org/wiki/Relative_atomic_mass). 

The atomic mass $m_a$ is the mass of a particular isotope of an atom, and if it is in 'u' (unified atomic mass units) it is relative to 1/12 of the mass of an atom of carbon-12. 

So, to get $A_r$ you take a weighted average of $m_a$ in 'u' for each isotope.

Note also that the standard atomic weight is the relative atomic mass of an element using natural abundance on Earth (http://en.wikipedia.org/wiki/Relative_atomic_mass).