quadratic twists of $P_{\mathbb{Q}}^1$

Question

I have seen somewhere else that if $C$ is a conic in $P_{\mathbb{Q}}^2$,then it is a quadratic twist of $P_{\mathbb{Q}}^1$.

So I want to know how to define the quadratic twist of $P_{\mathbb{Q}}^2$(maybe for a general variety?) and why is this curve isomorphic to $C$(as an algebraic variety)?

Thanks!