I am a little bit of a learner in number theory/arithmetic geometry, so feel free to redirect this question to math.stackexchange if necessary.
The question is the following: I know the Hasse principle for $\mathbb{Q}$: it holds if an alg. variety has a point over $\mathbb{Q}$ if and only if it has it over the p-adics $\mathbb{Q}_p$ and over the reals.
Recently, I have seen written somewhere "Hasse principle for a number field". Which is the definition in that case?
Over any global field the Hasse principle is that a point exists iff there's a point over every completion. Ostrowski's theorem says that the only completions of the rationals are the p-adics and the reals, so you recover the principle you know. In general there will be finitely many places over each rational prime $p$ and finitely many infinite places.