All physical objects have ‘mass’. The mass is (largely) a consequence of the additions of the masses of the atomic and subatomic particles from which the objects are comprised. Theorists currently support the idea of mass-creation via the Higgs Mechanism. See Wikipedia
But Higgs theory doesn't explain how or why subatomic particles themselves have wildly differing masses. For example, the 'up' and 'down' quarks which make up protons and neutrons, have different masses (by a factor of 2) - and the 'top' quark has a measured mass approaching that of some entire atoms.
There is also no explanation as to why the masses of some particles are “100 quadrillion times smaller” than theory would suggest. Source:'The Higgs mass mystery' New Scientist, Nov. 2015.
Further, as well as the current theories that put 'ordinary' matter as accounting for just 5% (or so) of the mass of the universe 1), Higgs theory only accounts for around 2% of this 5%. Put another way, this means that only 0.1% of the mass of 'ordinary matter' is currently explained.
“these current-quark masses leave 98% of the mass of the atom unexplained.” source
A further complication is that the mass of the Higgs particle itself has yet to be explained. “[…] the problem cannot even be formulated in the strict context of the Standard Model”
Note: A new (2016) (US/Japan/China) study details theoretical and experimental research on supercooled rubidium which claims to demonstrate 'Negative Mass'.
Theoretically, a 'Negative Mass' object would move towards a force attempting to accelerate it, or would fall upwards in a gravitational field (etc etc).
“Negative mass can be realized in quantum systems by engineering the dispersion relation. A powerful method is provided by spin-orbit coupling, which is currently at the center of intense research efforts. Here we measure an expanding spin-orbit coupled Bose-Einstein condensate whose dispersion features a region of negative mass.”