Cannot understand why positive semi-definite matrices are interesting in machine learning

Question

Deep Learning book states the following fact

and I cannot understand why they are interesting what what's so special about $x^{T} A x \geq 0$ and what are $x$ and $A$ in machine learning setting and what $x^{T} A x \geq 0$ implies.

Also why $x^{T} A x \geq 0$ on the first place? For some reason got really confused...

Thank you