Matter is hard to define but easy to explain. Every object you know is made of matter. That matter, in turn, is composed of atoms and molecules; atoms are tiny bits of matter that combine with other atoms to make molecules, which are slightly larger bits of matter. In turn, atoms are made of protons, neutrons, and electrons. (Atoms are also composed of particles known as quarks and leptons, as protons and neutrons are made up of quarks and electrons are a type of lepton.)
The most experience laypeople often have with the concept of matter is in the "three states of matter." Matter can be solid (having a defined shape and volume), liquid (having no particular shape but a defined volume), or gaseous (having neither defined shape nor volume). This is not, however, the definition of matter.
Matter is related to energy. There are 9.0 x 1020 ergs of energy in every gram of matter.
Idealists dispute the existence of matter, saying instead that what seems like matter is really an "idea" of some sort.