Fibonacci series and golden ratio

I was wondering if someone could explain the fibonacci series and golden ratio to me, I'm very curious, but I don't have that much experience in math as a high school trig student.

Personally, I liked that $$$\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}} = \dfrac{1}{1 + \dfrac{1}{1+\frac{1}{1+...}}} = \phi$$$ or the golden ratio. Also, note (since phi is defined by those recursive sequences) that $$\frac{1}{\phi} = \phi -1$$