pH stands for "potential hydrogen", originally "power of hydrogen". In simplest terms, it is a logarithmic scale (based on powers of 10, similar to the Richter scale for earthquakes, the decibel scale for sound and signal levels, and others that seek to measure very large differences with smaller numbers) of how many free protons there are in an aqueous solution of a substance, which is in turn a measure of that solution's acidity or basicity.
The number is the absolute value of the exponent of this measurement, so if, for instance, a substance has a pH of 7, that means there's one free proton per 10 million water molecules (1/10,000,000 = 10-7). A pH of one, or less, means there's at least one proton for every 10 water molecules, while a pH of 14 means there is one proton or fewer per 100 trillion water molecules.
Protons, or H+ ions, are attracted to the negative charge region of a water molecule. Water's bent molecular structure, with the two hydrogens closer to one side and oxygen's extra lone pairs on the other, effectively makes each molecule a tiny magnet, which is the foundation for many important characteristics of water, like why ice floats and why some things dissolve in water while others don't. Anyway, the hydrogen loosely "binds" to one of those lone pairs, forming hydronium (H3O+).
There is a natural equilibrium of one hydronium ion per 10 million molecules. At this point, pure water (again being a polar solvent) will actually dissolve itself to a small degree, producing one hydronium and one hydroxide per 10 million. At this pH of 7, the opposite measure, pOH (power of hydroxide) is also 7, balancing the reactivity of the water.
Below a pH of 7, the positively-charged hydronium atoms come to dominate this equilibrium, decreasing the number of hydroxide ions and looking for a substance to which to donate its protons to reach a balance again. This solution, which is now a "proton donor", is one of the classical definitions of an acid (the Lowry-Bronsted definition). Ideally, it will donate its proton to a substance that needs one as badly as the acid needs to get rid of it, but if the acid has enough hydronium, it will stick those hydrogens wherever they'll fit into any other substance's molecules.
Above pH 7, the negatively-charged hydroxide ions reduce the hydronium and dominate equilibrium. The solution is now a "base", a "proton acceptor", and is looking for something from which to take protons. The ideal substance to do so is a proton donor - an acid - but similar to an acid, a base, if strong enough, will take hydrogen atoms from anywhere they're not nailed down.
Now, not all substances which are acidic have a free proton to donate (and so not all of them increase the amount of hydronium ions in water), and not all substances which are basic have a hydroxide anion, nor do they accept free protons into their structure. Therefore, there is another definition, the Lewis definition, stating that an acid is a substance that accepts a "lone pair" of electrons from another substance (a pair of electrons not forming a bond between atoms of the molecule) to equalize its own molecular charge, and a base is a substance that readily donates an electron pair. Regardless of the definition or the exact mechanism, these acids and bases have similar observed properties, and so even if explicit exchange of hydronium atoms isn't involved, the substances still indicate a pH using various testing methods. This prompted the change in name from "power of hydrogen" to "potential hydrogen", in effect making the measurement an "equivalent", applying to both Lewis acids and bases and Bronsted-Lowry acids and bases.