Volume
Volume is the amount of space an object occupies.
This seems a simple concept, and it is for basic, solid objects. For a solid cube, for example, multiply length by width by height to obtain the volume. But what is the volume of a gas, which can easily expand and contract? Except for idealized mathematical models, the volume of an object depends on the composition of that object, and the environment around it. Scientists have spent significant effort to formulate how such quantities depend on their environmental quantities like pressure and temperature. The volume of a solid, for example, typically expands slightly as the temperature increases, but in general they are mostly resistant to changes in their shape or volume. A liquid like water will take the shape of whatever container it is poured into, but its volume stays the same. The gas in a balloon will take whatever shape the balloon has, and the volume will shrink if external pressure is applied to the balloon. (Ideal gases, in fact, obey Boyle's Law, which states that the volume is inversely proportional to the pressure, at constant temperature.)
Greek mathematicians, especially Archimedes, were among the first to derive formulas for the volumes of common shapes. It was not until the invention of calculus by Isaac Newton and Gottfried Wilhelm Leibniz in the seventeenth century that the volume for an arbitrary shape could be calculated. One can show, for example, that for an object of a certain volume, a sphere has the smallest ratio of surface area to volume. This explains why a cat curls up into a ball when it sleeps--it is trying to make itself like a sphere, so its surface area is as small as possible in order to minimize heat loss through its skin.
For a solid of an unusual or arbitrary shape, one could determine its volume by, for example, placing it in a full tub of water and measuring how much water spills over the edge. Lowering himself into a full tub, Archimedes is said to have been inspired to formulate his law of buoyancy.
Why does matter have volume? That is, since matter is composed of atoms which have a great deal of empty space inside them and between themselves, why doesn't the atoms just collapse into a tiny, solid clump that has an extremely small volume? Because the elements of the atoms--the electrons, protons, and neutrons--have quantities like electrons, electric charge that makes them repel one another if they get too close.
The volumes of some typical objects can vary enormously. For example, the human body has a volume of roughly 0.1 cubic meters, which is 100 liters or about 26 gallons. A grown elephant's volume is about 6 cubic meters (1,500 gallons!), and the volume of the earth is about 1021 cubic meters. By contrast, the volume of a hydrogen atom is far, far smaller, about 10-30 cubic meters.
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