Meaning of $x^T A x$

I've seen the term $x^T A x$, where $A$ is a square and usually symmetric matrix, come up in a bunch of different areas of linear algebra. Places I've seen it include defining the Raleigh quotient, defining positive/negative semi-definite matrices, and in the derivation of PCA. I've also seen it sometimes referred to as describing a quadratic form.

Is there some general definition/ intuitive description of what $x^T A x$ means with respect to a vector and a matrix?

My sort of vague understanding is that it describes how a vector is changed under a linear transformation defined by $A$ (for example if $A$ causes x to rotate 90 $^\circ$ then $x^T A x = 0$) but I can't seem to come up with a more precise or insightful description of $x^T A x$, and I'm surprised how little I could find online considering how often I see this term come up.

$$Ax$$ represents the vector $x$ after it has undergone the linear transform described by $A$ and

$$x^TAx$$ is the dot product of the original and the transformed vectors. This dot product is zero when the two vectors are orthogonal, and maximized/minimized when they are parallel/antiparallel.

If $x$ is an Eigenvector of $A$,

$$x^TAx=\lambda x^Tx=\lambda\|x\|^2.$$

More generally, $x$ is a linear combination of Eigenvectors $ue_1+ve_2+\cdots we_n$ and

$$x^TAx=\lambda_1u^2+\lambda_2v^2+\cdots\lambda_nw^2$$ is a canonical quadratic form. Depending on the signs of the lambdas, the equation $x^TAx=1$ describes an ellipsoid or some kind of hyperboloid.