This is an exercise from Chang & Keisler, specifically, exercise 1.3.1, though I'd also like some information on the (I think) related exercises 1.3.6 and 1.3.8. It's as the title of this question states:
Let $\alpha$ be any cardinal. There are at most $2^{\alpha \cup ||\mathscr{L}||}$ nonisomorphic models for $\mathscr{L}$ of power $\alpha$
where $||\mathscr{L}||$ denotes the cardinality of the language (which, as usual, is at least $\omega$). I admit that I'm a bit at loss as to how to even begin the proof. I suppose part of the problem is that we're considering here the number of models for a given langauge, instead of for a given theory, so I'm a bit unsure of which ifnormation is relevant for that. In the first part of the exercise, you're asked to prove that the isomorphism relation between two structures is an equivalence relation (a rather trivial proof); so I think this fact should be used in the proof somehow, yet I can't quite see how yet. If anyone could give me a hint, I'd be very grateful.
Note that every $L$-structure $M$ of size $\kappa$ is isomorphic to an $L$-structure with domain $\kappa$. Indeed, given a bijection $f\colon M\to \kappa$, it's easy to see that $\kappa$ can be made into the domain of an $L$-structure in a unique way so that $f$ is an isomorphism.
Hence the number of distinct $L$-structures up to isomorphism is less than or equal to the number of distinct $L$-structures with domain $\kappa$.
The rest is just a matter of counting how many choices you have to make to describe a structure with domain $\kappa$. Since you only asked for a hint, I'll leave the details to you.