Lambda calculus is a theoretical programming language (with same expressive power as a Turing Machine). It is interesting because there is no state, and it consists only of two operators: . and λ.
If you want to learn more, visit the Wikipedia article http://en.wikipedia.org/wiki/Lambda_calculus as this will be only a short introduction.
So, since we have no state, we have no variables. We only have constants. Numbers are defined like:
1 := λf x. f x
2 := λf x. f (f x)
In other words, we can define 1 as f(x), and 2 as f(f(x)) (we get to define f and x ourselves).
Functions in lambda fall in two categories: abstraction and application.
Abstraction is when we define a function and apply no arguments to it.
Application is when we apply arguments to some abstraction (already defined function).
SUCC (successor) example:
SUCC is defined as
SUCC := λnfx. f (n f x)
So, doing SUCC 1 we get
(λnfx. f (n f x))(λf x. f x) =
λfx = f ((λf x. f x) f x) = f (f x) = 2
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