There is work we can do here to clarify the contexts, work that will pay dividends even though it might imply clean lines and sharp corners where there are none. First, note that I am helping myself to the term 'context' here without much fanfare. I take it to be a repeatable type of event that can include people. Second, an argument context will be a type of context, i.e., it will be abstract and repeatable. As such, we will specify it by describing features that are realized in different ways and to different degrees by token, i.e., specific, contexts. Third, and unsurprisingly, argument contexts are contexts where you typically find argumentative discourse.
What follows is a short and rather random list of typical argument contexts, with a bit of description about each. Argument contexts come in many shapes and sizes, often emerging from the way they are described. I have tried to stick close to rather obvious cases in compiling this list. If you find yourself in one of these, it is likely that you will be surrounded by argument. So, in no particular order, the contexts:
Quarrel: rather unstructured events fed by emotion that can lead to name-calling, shouting, and violence; while quarrels will always involve the activity of argument, it is not too uncommon for them to lack arguments understood as rationales.
Advertisement: attempts to sway or mold opinion in favor of a product. Advertisers want to sell you things. They want you to conclude, "OK, I'm buying that." The reasons are typically shown in the ads---because buying it will make you stronger, sexier, smarter, more popular, etc. Many parts of the arguments here are implicit, that is, unstated in the ads. But each ad is almost always an argument.
Debate: highly structured events where victory is predicated on advancing the rationale that most impresses the judges; it is rare to find a debate that lacks arguments, although it isn't rare to find a debate that lacks good arguments.
Persuasive Discussion: often rather unstructured conversations in which participants attempt to convince each other to embrace certain beliefs (e.g., discussions about politics, religion, morality, the AP basketball top 25, etc.); since these are individuated by virtue of their persuasive character, such discussions usually do not lack argumentation of one sort or another.
Inquiry: confronted by a problem, e.g., ignorance of a desired fact, uncertainty about a decision, etc., an individual or individuals seek a resolution by examining what can be said for potential solutions; while it is typically guided by the evaluation of solutions, and so by the construction of arguments for or against them, inquiry can at times yield more head scratching than headway.
Opinioneering: advancement of an opinion, such as in a Letter to the Editor, a column or Op/Ed piece, or a sermon, that is not immediately dialogical in nature; these can lack arguments, in the event that the opinion is merely asserted without supporting reasons, perhaps over and over again in only slightly different ways.
Negotiation: driven by a sustained desire for agreement in the context of a difference of interests, this is marked more by compromise than persuasive discussion; successful negotiation will typically involve argumentation adduced in favor of initial positions and then a series of compromises.
Action Planning and Pursuit: marked by the need for action, whether by an individual or group; this process involves the presentation and subsequent evaluation of practical options, followed by comparative assessment of those that emerge as viable; this may not be argumentative in character, such as when options are "rammed through".
Learning Environment: classrooms, offices, and administration lawns supply space for the give-and-take of teacher-student interaction; in such environments, teachers often present arguments in the course of lecturing, and arguments are often advanced in discussion, group work, etc.; however, these can be non-argumentative, especially when they involve simple information transfer from one brain to other brains.