The mass of an atom is referred to as atomic mass. The mass of a molecule may also be referred to as atomic mass but may also be referred to as molecular mass. In either case, atomic mass refers to the average mass of a pure sample.
Mass is a basic property of matter and is made of atoms. Atoms are made of electrons, protons, and neutrons. Protons and neutrons make up the majority of the mass in an atom. Atoms may vary in the number of neutrons they have (called isotopes), and therefore the exact atomic mass may vary from atom to atom.
The mass of a single atom is called relative isotopic mass and is scaled based on carbon-12, which was assigned a value of 12 due to its most abundant composition having six protons and six neutrons. Unified atomic mass units are the label used for describing atomic mass and are always close to whole number, except for carbon-12 which is a whole number due to it defining the scale. Protons and neutrons have similar mass and because they make up the majority of mass in an atom, it is common to use integer mass instead of atomic mass to describe the mass of samples. Integer mass has the label "daltons" or "u" for units.