What's an example of an unsatisfiable 3-CNF formula? I'm trying to wrap my head around an NP-completeness proof which seem to revolve around SAT/3CNF-SAT.

Maybe it's the late hour but I'm afraid I can't think of a 3CNF formula that cannot be satisfied (I'm probably missing something obvious).

Can you give me an example for such formula?
Technically, you can write $x\wedge \neg x$ in 3-CNF as $(x\vee x\vee x)\wedge (\neg x\vee \neg x\vee \neg x)$, but you probably want a "real" example.

In that case, a 3CNF formula needs at least 3 variables. Since each clause rules out exactly one assignment, that means you need at least $2^3=8$ clauses in order to have a non-satisfiable formula. Indeed, the simplest one is:

$$(x\vee y\vee z)\wedge (x\vee y\vee \neg z)\wedge (x\vee \neg y\vee z)\wedge(x\vee \neg y\vee \neg z)\wedge(\neg x\vee y\vee z)\wedge(\neg x\vee y\vee \neg z)\wedge(\neg x\vee \neg y\vee z)\wedge(\neg x\vee \neg y\vee \neg z)$$
It is not hard to see that this formula is unsatsifiable.