The atomic mass of an element is the mass of one atom of that element. They can be measured in a fairly accurate manner with the help of mass spectrometry. Via experimental means, the mass of one atom can be measured relative to the mass of another. Since hydrogen is the first element (its atomic number is 1), the relative atomic mass of hydrogen is 1. The unit of atomic mass is the atomic mass unit, which is defined as the mass equal to 1/12 of the mass of a single carbon-12 atom. It is approximately equal to 1.66*10-24 grams.
Average atomic mass takes into account the isotopic abundance (relative to each other found in the Earth). For example, the most abundant isotope of carbon is carbon-12, which has a relative abundance of 98.89%. The atomic masses of carbon-13 and carbon-14 are 13.003 amu and 14.003 amu respectively. The average atomic mass of carbon is 12.011 amu.