As the name suggests, molecular mass is the mass of a molecule. You will hear terms such as molecular weight, formula weight, formula mass, and molar mass used in conjunction with molecular mass. There are very slight differences, but the general concept is the same for all. The expectation is that every molecule that is made up of the same atoms and the same bonds will have the same mass, and this mass can be measured or predicted mathematically.
Molecular masses are given units of amu, or atomic mass units. This unit is defined such that one atom of carbon-12 (a carbon atom with six protons and 6 neutrons) has a mass of exactly 12 amu. As a result, the amu is fairly close to the mass of a neutron. The advantage of using the amu rather than the mass of a neutron is that the mass of a neutron varies depending on the nucleus it is in. (If you remember Einstein’s E=mc2, this is one place where it matters - the formation of atomic nuclei results is a characteristic loss of mass.)
The amu is a very small unit, and not generally applicable to anything life-sized. Thankfully, it is interchangeable with another unit - grams per mole (g/mol). Grams, are a basic unit of mass, while the mole is a very large number. The mole is the number of carbon-12 atoms that have a total mass of exactly twelve grams. (Just like the amu, the mole is a defined quantity.) When a molecular mass is presented as the mass of a mole rather than the mass of an individual molecule, it is called the molar mass. For more on moles, you might consult the article “How Big is a Mole?”
Molecular masses can be estimated by consulting the periodic table. Most periodic tables provide an average atomic mass for each element. (An average is used because different isotopes of an element have different masses.) The chemical formula of a molecule describes how many of each atom are present, and then it is a simple matter of adding together the appropriate masses from the periodic table. This is the formula mass mentioned earlier. The formula mass of a molecule does not reflect the actual molecular mass of any molecule, but it serves as an approximation that is good enough for most applications.
True molecular masses are a little more difficult to obtain. The mass of a molecule is not the sum of the masses of its constituent atoms. When atoms bond to form molecules, a small amount of energy is lost, resulting in a minuscule loss of mass (E=mc2 again). For this reason, even two chemicals with the same chemical formula may have unique masses if they have different bonds. The difference is very small and may not be apparent until the fourth or fifth decimal place, which is why it can be ignored in everyday calculations.
Computational software exists that allows chemists to make good predictions about molecular behavior, including mass, however, the only way to truly measure the mass of a molecule is using mass spectrometry. The mass spectrometer is a highly precise analytical instrument which uses electric or magnetic fields to exert forces on a charged molecule. The ratio of the mass of the molecule to the charge (typically the charge equivalent to one electron) on it governs how the molecule moves, allowing determination of masses at a small fraction of an amu. If the instrument has adequate sensitivity, a mass spectrometer can distinguish between two molecules which have the same chemical formula but different structures by the small difference in their masses.
Mentioned at the beginning of the article were two other terms - molecular weight and formula weight. In common usage, molecular weight means the same as molecular mass, and formula weight means the same as formula mass. Because they are in common usage, any chemist will know what you mean if you use them. From a purely technical standpoint however, anything expressed in amu or g/mol is a mass, and not a weight. Weights require units of force, like Newtons or pounds. (Weight is dependent on gravity, whereas mass is not.)