What are 'contexts' actually called? Consider the following argument by contradiction.
\begin{array}{|l}
\mbox{We wish to deduce A.} \\ 
{\begin{array}{|l}
\mbox{Suppose not A.} \\
\hline \\
\mbox{Then B. Thus C. Therefore, contradiction. }
\end{array} } \\
\mbox{Thus, A.}
\end{array}
So to actually perform the argument by contradiction, we had to open a new ‘context’ wherein additional assumptions hold. What are these ‘contexts’ actually called? And which deductive systems actually utilize them?
 A: In general, natural deduction systems are most commonly associated with what you call "contexts". People call them different things, but I often hear "assumptions" or "suppositions" used, and the proofs that follow are "subproofs". When you move outside the assumption, you are discharging the assumption. An example of a natural deduction system is the Fitch system, which has resemblance to what you present. Tree proofs are also generally natural deduction systems.
Two classes of systems which don't utilize this idea are Hilbert axiomatic systems and sequent calculi. They have different ways of handling the idea of making assumptions. Hilbert systems, for instance, just force your assumptions to end up in the antecedents of conditionals one way or another, which makes the proofs much more tedious.
A: Natural deductive systems often use such a context often for a few different rules.  Say we want to prove that Cpr (p implies r) in some proof.  Then we'll open such a new context, take p as a supposition and show that r follows.  Then we infer Cpr outside of the context.  Also, you can open a new context supposing p, show that a contradiction follows, and then infer Np (the logical negation of p) outside of the context.  The new contexts I've seen called "subproofs".
A: The other answers have discussed subproofs in natural deduction systems for conditional proof.  There are also natural deduction systems that have subproofs for other types of contexts, such as necessity introduction in modal logic.  For instance, axiomatic modal logics typically have the inference rule $$\vdash \phi / \vdash \Box\phi$$ which states that if $\phi$ is a theorem then so is $\Box\phi$, and the axiom $$\Box(\phi \to \psi) \to (\Box\phi \to \Box\psi)$$ which says that “if $\phi$ necessarily implies $\psi$, then if $\phi$ is necessary, so is $\psi$.”  In natural deduction systems for modal logic, axioms can be captured by ‘modal contexts’ or modal subproofs.  For instance, in Introductory Modal Logic, Konyndyk uses, in addition to conditional subproofs of the type described in the question, modal subproofs of the form shown in the following:
\begin{array}{l}
{\begin{array}{|ll}
\Box P  \\
\hline
{\begin{array}{|ll}
\Box Q \\
\hline
{\Box\begin{array}{|ll}
P & \mbox{reit} \\
Q & \mbox{reit} \\
P \land Q & \mbox{$\land$ intro}
\end{array}} \\
\Box(P \land Q) & \Box\mbox{ intro} \\
\end{array}} \\
\Box Q \to \Box(P \land Q) & \to\mbox{ intro}
\end{array}} \\
\Box P \to (\Box Q \to \Box(P \land Q)) & \to\mbox{ intro}
\end{array}
The idea is that necessary things can be reiterated into a modal subproof, but they lose the leading $\Box$ when they move in.  Different modal logics generate different constraints on what can be reiterated in, and whether or not the modality is dropped upon reiteration.  These ‘contexts’ are still called (modal) subproofs.
Another point of interest comes up in first-order logic, where a subproof may introduce a context in which a something is taken as a term denoting an arbitrary individual (for universal introduction) or a witness individual with a particular property (for existential elimination).  These ‘contexts’ are also typically referred to as subproofs.
