$$\Delta, A \vdash B \implies \Delta \vdash A \to B$$
I've seen this set $\Delta$ referred to as a set of "assumptions", a set of "hypotheses", a set of "axioms", a set of "non-logical axioms", a set of "postulates", a set of "schemata", etc.
I'm not entirely sure what this means formally, if these are all actually saying the same thing -- but if that's the case and an "assumption" is the same as a "non-logical axiom" I'm not sure how this differs from a regular "axiom" that we normally build right into the system to begin with.
Especially when we define a proof as a sequence of lines $\varphi_1, \varphi_2, \varphi_3, ..., \varphi_n$, where "each line is either an axiom or theorem", I can't tell if this includes non-logical axioms / assumptions, or if we have some special name for the results acquired through modus ponens (are these theorems?). I know a "theorem" for example is composed of axioms and other theorems via logical connectives but I don't know if this includes non-logical axioms or modus ponens results.
Furthermore I don't really know where these assumptions/hypotheses/non-logical axioms "come from" -- if they're invoked out of thin air $\vdash \phi$, or if they may not necessarily be sound/true or what have you.
Can anyone shed some light on this? What's contained in $\Delta$ exactly? Where do they come from exactly? What is each line of a proof allowed to be exactly?