To achieve recursion in lambda calculus, a fixed-point function is required. The fixed-point function is generally referred to as Y, and must by definition satisfy Yf=f(Yf).

The function used for Y is λf.(λg.f(gg))(λg.f(gg)). Yf can be beta reduced to (λg.f(gg))(λg.f(gg)), which in turn can be beta reduced to f(λg.f(gg))(λg.f(gg)), satisfying Yf=f(Yf).

Using Y, a function has access to a bound copy of itself.
If lambda expressions were named, you might want to write f=λx_{1}...x_{n}E, where E is some expresion refering to f.
With the Y combinator, the function on the right becomes Yλfx_{1}...x_{n}E.