pH is a mathematical formula that stands or the negative of the base 10 logarithm of the molar concentration of hydrogen ions. pH = -log_10 ( [H+] ) where [H+] is shorthand for the concentration of H+.
Example, the concentration of a solution has 5 * 10^-8 moles of H+ per liter (that would be 0.000 000 050 moles per liter), it's pH is - log_10 ( 5*10^-8 ) = - ( - 7.3 ) = 7.3
The amount of H+ ions determines the acidity or alkalinity of a solution. Acids give up H+ ions, bases take up H+ ions. For example, Hydrochloric acid, HCl readily disassociates into H+ and Cl-, it gives up H+ readily. Sodium Hydroxide is a base. Sodium Hydroxide is NaOH, which readily disassociates into Na+ and OH-. The OH- and the H+ combine to give back H2O, so sodium hydroxide accepts H+, and is considered a base. Real life can be much more complicated than this, there are compounds that can both give up and accept H+'s, and a compound's acidity and alkalinity isn't always just a function of whether it gives up or accepts H+'s, but this is a good enough definition to handle most common cases.
There isn't only pH, though that is by and far the most commonly seen one. You can also have a pOH of a solution, and a lot of times disassociation constants are reported as pKa values. It really is just a way of writing a number that has a unit and a few numbers after a decimal rather than writing the number in scientific notation.