Short answers, pulled from the story.
A variable in mathematics is a symbol, typically a letter, that refers to an unspecified mathematical object. The variable is said to represent or denote that object, and any valid candidate for the object is the value of the variable. The values are usually of the same kind, often numbers, and may form a set such as the set of real numbers.
Rene Descartes introduced the convention in 1637 of representing unknowns in equations with letters from the end of the alphabet and knowns with letters from the beginning. This differed from Francois Viete, who at the end of the 16th century used consonants for known values and vowels for unknowns. Descartes' convention is still commonly in use.
An unknown is a variable in an equation that has to be solved for, while a parameter is a quantity that forms part of a problem's input and stays constant during its solution. An indeterminate is a symbol in a polynomial or formal power series that is formally a constant in the polynomial ring. A parameter remains fixed during a problem, an unknown must be determined, and an indeterminate is treated as a special kind of variable because of the link between polynomials and their functions.
The earliest known uses of an unknown quantity date back at least to the Ancient Egyptians with the Moscow Mathematical Papyrus, around 1500 BC, which described problems with unknowns rhetorically as the Aha problems. Around the same period in Mesopotamia, the Old Babylonian mathematics of roughly 2000 BC to 1500 BC studied quadratic and cubic equations.
A dependent variable is a variable that is implicitly a function of another variable or several other variables, while an independent variable is one that is not dependent. Whether a variable is dependent or independent often depends on the point of view and is not intrinsic. In physics, quantities such as pressure and temperature are treated as variables dependent on time.
Karl Weierstrass shaped the modern notion in the second half of the 19th century by replacing the intuitive idea of a limit with a formal definition, in a formula where none of the variables is considered as varying. This static formulation led to the modern variable as a symbol representing a mathematical object that is either unknown or may be replaced by any element of a given set. Earlier, Isaac Newton, Gottfried Wilhelm Leibniz, and Leonhard Euler had tied the word variable to the arguments and values of functions.