Boyle's Law states that at a fixed temperature and within a closed system, the volume of a fixed quantity of a gas is inversely proportional to the pressure it exerts: PV = k, where k is a constant. For example, the volume of air in a balloon exerts enough pressure to stretch the rubber but break that rubber, and suddenly the the artificially high pressure of the balloon is completely gone because the air it had contained under pressure has rushed out and filled the greater volume of the surrounding space. For another example, in pressing down a tightly-sealed plunger, the pressure (difficulty) increases as the volume of the air under the plunger decreases. Release the plunger, and it will rise (increasing the volume of the air within its enclosed space) to a point where the pressure of the sealed volume matches the air pressure outside.
The British chemist Robert Boyle understood this outcome to be a function of gas as a collection of 'corpuscles', or tiny particles. The corpuscular theory of matter, better known as atomism, actually goes back in Western thought as far as the late 5th century BCE, when Leucippus and his student Democritus first introduced the philosophical idea that all physical objects consist of different arrangements of indivisible particles and void, and that temperature exists only as an organism's sensation based on the manner of arrangement and scattering of these atoms. In fact, this is the source of the word atom, from the Greek "a-tomos", or not cuttable. In many ways this concept is not so very different from the modern molecular understanding of the nature of matter. However, both Plato and Aristotle objected to the cold randomness of the atomist concept. Plato felt that all matter was the reflection of perfect forms, which would logically exist of perfect geometry. Aristotle, in contrast, thought that all matter was composed of the four continuous elements of earth, air, fire, and water. This latter view took hold of the public consciousness, and was never entirely relinquished until the dawn of the scientific method during the Renaissance.
In 1612, following the 'scientific method' advocated by Francis Bacon whereby the natural philosopher should replace deductive syllogism (using reason to extrapolate from existing statements) with inductive reasoning (using observed fact to deduce axiom and then law), Gallileo Galilei theorised that all phenomena except sound resulted from "matter in motion". This is the basis of the corpuscular theory of matter, which is essential for understanding why gas pressure might increase with decreased volume: the particles are simply pushed closer together, beyond their 'natural' mutual distances and consequently into increased collisions with each other.
When he published the law linking volume and pressure, Boyle himself referred to it as Richard Towneley's hypothesis. In 1661, Towneley and his friend Henry Power had used a barometer to measure air pressure at different altitudes, and thereby deduced a relationship between air pressure and air density. The resulting book, "Experimental Philosophy", was not published until 1663, but Towneley had discussed the experiments with Boyle and also shown him an early draft of the book. Boyle tested this hypothesis within a closed chamber apparatus which made use of vacuum pumps, publishing the results in 1662: and thus the resulting law bears his name.
The apparatus itself was built by his assistant Robert Hooke, who is also held to have been a better mathematician than Boyle, and who thus might well have done the lion's share of the data crunching. Hooke later developed what is still known as Hooke's law of elasticity, which he summarised as "Ut tension, sic vis", as the extension, so the force. Clearly the two lines of reasoning are tightly interrelated.
Boyle's law is sometimes known as the Boyle-Mariotte law, to acknowledge the 1676 contributions of the French physicist Edme Mariotte. There is some question as to whether Mariotte did in fact discover the principle independently of Boyle's writings.
It is important to appreciate that Boyle's law is a real-world approximation best suited for anticipating the effects of relatively small change on a closed system, rather than an absolute holding true across all possible pressures at all possible temperatures. At greater pressures or higher temperatures, the assumed simple collision and bounce-back begins to be replaced by a model where some molecules continue to be mutually repelled, while others begin to interact chemically with each other as their activation energy is exceeded. Similarly, at higher complexities of molecules, the clean assumption of simple collision theory, that molecules approximate a spherical shape equally able to react in all directions, breaks down.