Show that Halting problem $(\mathsf{HP\text{}})$ is $\mathsf{NP\text{-}Hard}$
Given a SAT formula $\varphi$, construct a Turing machine which iterates over all truth assignments for $\varphi$, and so determines if $\varphi$ is satisfiable or not. If it is, it halts. If it isn't, it doesn't halt (gets into an infinite loop).

As is apparent, in the same way we can show that the halting problem is coNP-hard (by reversing the decision rule at the end). Indeed, in the very same way we can show that the halting problem is hard for many other classes – in fact, for all classes defined by time or space constraints (deterministic, non-deterministic or randomized).

NP is the "length- and time-bounded" version of the halting problem, or rather, of $\Sigma_1$. The class $\Sigma_1$ consists of all predicates $P$ such that
$$ P(x) \leftrightarrow \exists y  \, \Pi(x,y), $$
where $\Pi$ is a computable predicate. NP is defined in a very similar way, by adding two restrictions:


$y$ has to be polynomially bounded (in $x$).
$\Pi$ has to be computable in polynomial time.


If you like this sort of thing, there is an entire field dedicated to this point of view, called Bounded Arithmetic.