I am having a really hard time understanding what a dual basis is.

I understand that

is the vector space of linear transformations from a vector space V onto its field of scalars F, also known as linear functionals.

I get this. What I am struggling on is how to find the dual basis and why it is defined the way it is.

We call the ordered basis

of

that satisfies

,

, the dual basis of

No: if is a basis of V, then as defined above is THE dual basis of X in , or wrt the basis X. Its importance resides,among other possible things, on the fact that it is one simple way to show that when we're dealing with finite dimensional vector spaces, then
The book only gave on example using

. I followed what they were doing but it doesn't help me understand the dual basis any better.

Based on the way we set up bases for any other vector space, I quess I figured that the dual basis would somehow be a set of linear functionals such that every linear functional could be written as a linear combination of these linear functionals.

Exactly, but NOT only: every element in V* indeed can be expressed as a lin. comb. of a dual basis, WHERE the elements of this basis get very simple and explicit values on some given basis of V.
I don't know, I am lost. Can someone explain this to me better or possibly refer me to a good website to read in more depth about dual spaces and dual bases?