Weight
In modern scientific usage, weight and mass are fundamentally different quantities: mass is an intrinsic property of matter, whereas weight is a "force" that results from the action of gravity on matter: it measures how strongly the force of gravity pulls on that matter. However, in most practical everyday situations the word "weight" is used when, strictly, "mass" is meant. For example, most people would say that an object "weighs one kilogram", even though the kilogram is a unit of mass.
Atomic mass
The atomic mass ("m") is the mass of an atom. Its unit is the unified atomic mass units ("abbr." u) where 1 unified atomic mass unit is defined as of the mass of a single carbon-12 atom, at rest. For atoms, the protons and neutrons of the nucleus account for nearly all of the total mass, and the atomic mass measured in u has nearly the same value as the mass number.