What does this function notation mean? My text tells me that the general term of a sequence can be looked at like a function:
$
f:\mathbb{N}\rightarrow \mathbb{R}
$
What does that mean translated into common english?

This specific case is called a sequence. The function $f$ takes in an input from the natural numbers (denoted $\mathbb N$), and gives an output in the real numbers (denoted $\mathbb R$). 
In general, this notation contains three parts: the function name, the domain, and the codomain. The function name (in your case $f$) is the same $f$ appearing in the notation $f(x)$. The domain is the set of inputs to your function (in your case the natural numbers $\mathbb N$). The codomain is the set of possible outputs that the function can give (in your case the real number $\mathbb R$).
