The definition of the relationship of eigenvectors and eigenvalues of a matrix is
$$Ax = \lambda x$$
where $x$ cannot be the $0$ vector.
When determining the positive definiteness of a matrix we do
$$x^TAx = \lambda x^Tx$$
and if $x^TAx > 0$, then it is a positive definite. My question is, why was $x^Tx$ added in the first place? Why is that needed? I understand that it is symmetric, but what is its relationship to $A$?
In the general definition, a positive definite matrix is a symmetric matrix $A$ for which the quadratic form $$x^TA x > 0,$$ for all nonzero vectors $x$. If $x^T Ax\geq 0$, then the matrix is positive semidefinite. Note that this holds for any $x$, not only for eigenvectors.
That is the definition. Using this you can find the rest of the results. For instance, if $A v = \lambda v$, then $\lambda$ is the eigenvalue associated with the eigenvector $v$. From the definition if positive definiteness, you have that $v^TAv > 0$, because $v$ is a nonzero vector, hence $$v^T Av = v^T(Av) = \lambda v^Tv = \lambda \|v\|^2 > 0$$ from which you see that positive definite matrices have positive eigenvalues. That is the relationship between eigenvalues and the quadratic form associated with a positive definite matrix.