Mass number, (relative) atomic mass, average mass
It’s pretty simple:


The mass number is the integer you get if you count (and add up) the neutrons and the protons of a given element. Thus, a hydrogen atom of the $\ce{^1_1H}$ isotope has a mass number of $1$ (only proton), $\ce{^12_6C}$ has $12$ ($6$ protons, $6$ neutrons) and $\ce{^81_35Br}$ has $81$ (of which $35$ are protons, the remaining $46$ neutrons). 
The atomic mass is what these atoms actually weigh in atomic mass units. For reasons that boil down to $E = mc^2$ (or so I believe) and the nonzero mass of an electron, this is not an integer except for one exception:


$\ce{^1H}$’s atomic mass is $1.007825032 1(4)$
$\ce{^12C}$’s atomic mass is exactly $12$. This is because $1~\mathrm{u}$ was defined as exactly $1/12$th of the mass of a carbon-12 atom.
$\ce{^81Br}$’s atomic mass is $80.916289(6)$

The average mass takes into account an elements different isotopes and their natural abundance and calculates an overall average. Thus, this is no longer defined on an isotopal basis but on an elemental one. The average masses of the elements discussed above are:


$\ce{H}$ has $1.00794 (7)$ (this is larger than the atomic mass of $\ce{^1H}$ due to the heavier isotopes.)
$\ce{C}$ has $12.0107 (8)$
$\ce{Br}$ has $79.904 (1)$ (this is lower than the atomic mass of $\ce{^81Br}$, because about half of the naturally occuring bromine atoms are $\ce{^79Br}$.)