What is matter? The answer depends on at what level we look at the building blocks of Nature.
(What is the matter? is another question altogether).
Matter is anything that has both mass and volume (occupies space). Like your body and mine, too.
Atoms and molecules definition
Matter is made up of atoms and molecules. The idea that matter is built from minuscule atoms colliding into "lumps" that give matter its attributes was miraculously figured out - just by thinking - by Leucippus and his student Democritus (460-370 BC) in ancient Greece.
Protons, neutrons and electrons definition
"Matter is made up of what atoms and molecules are made of, meaning anything made of positively charged protons, neutral neutrons, and negatively charged electrons."
Quarks and leptons definition
Matter is made of what its elements are made of, meaning everything that is composed of elementary fermions, namely quarks and leptons
Baryonic definitionBaryonic matter is the part of the universe that is made of baryons (including all atoms). Baryons are strongly interacting fermions, and so are subject to Fermi-Dirac statistics.
"This part of the universe does not include dark energy, dark matter, black holes or various forms of degenerate matter, such as compose white dwarf stars and neutron stars.
Microwave light seen by Wilkinson Microwave Anisotropy Probe (WMAP), suggests that only about 4.6% of that part of the universe within range of the best telescopes (that is, matter that may be visible because light could reach us from it), is made of baryionic matter.
About 23% is dark matter, and about 72% is dark energy."
"In the context of relativity, mass is not an additive quantity.
Thus, in relativity usually a more general view is taken that it is not mass, but the energy–momentum tensor that quantifies the amount of matter.
Matter therefore is anything that contributes to the energy–momentum of a system, that is, anything that is not purely gravity.
This view is commonly held in fields that deal with general relativity such as cosmology."
(Quoted texts in this posts are from the wikipedia article on matter.)