The idea of a function is fundamental to mathematics. A function is a relation between two sets (an "input" set and an "output" set) such that each input is related to exactly one output.
Each function has a domain of definition ("domain" for short). This is the class of numbers or other things to which the function can be meaningfully applied. An applicand of a function is any number in the function's domain. The result of applying a function to an applicand is called the value of the function for that applicand. The class of all values of a function for all its applicands is called the domain of values, or the range, of the function.
When discussing functions, it's helpful to denote the function by some name. You'll commonly see f, although other letters (e.g. g, h) are also common.
It can be useful to refer to a whole series or class of applicands at once; to do this, a variable is used. For example, f(x) = x² refers to a function whose output is the square of its input. Used in this way, a variable is called an argument of a function. While the above function takes only one argument, functions could of course take more than one argument.