Fibonacci Sequence explained to a noob

Each member of the Fibonacci sequence is the sum of the previous two members. There are two standard ways of starting the sequence - you might start with $0$ and $1$, or with $1$ and $1$.

Starting with $0$ and $1$, we have $0 + 1 = 1$; so the third member of our sequence is also $1$ and our sequence so far is $0,1,1$. $1+1 = 2$, so we now have $0,1,1,2$. $1+2=3$, so we have $0,1,1,2,3$. $2+3=5$, so we have $0,1,1,2,3,5$; and so on.

If you start with $1$ and $1$, you're just starting one step later, so you get $1,1,2,3,5,\ldots$, which is the version you've seen. Whether the Fibonacci sequence is "supposed" to start with $0$ or with $1$ is really just a matter of taste.