Surface level
Small talk
The conversation doesn't matter very much for either participant. Is mainly there to prevent an awkward silence or to pass the time until it's socially acceptable to separate. Nothing important is learnt or shared.
Conference
The word conference comes from the Latin word conferre, which means to compare. By conference I mean any conversation which is about sharing information that one party doesn't have, but not much more than that. The sharing can go both ways. Examples of conferences are: tips on things to buy between consumers, tips on rules and regulations between home owners, or system knowledge between coworkers. Canned 'fun facts' may also belong here. Conferences often happen between people who are getting to know each other. It's usually necessary sooner or later to go through the ways the two are different, in order to establish what feels like a real relationship. This can be if they are from different places, come from different cultures, or are different ages. However, conference is not a substitute for shared reality conversations (see below). A failure mode of romantic relationships is to confuse 'conference type' conversation topics with your new partner for having a lot to talk about. Conference is a finite resource, sooner or later you simply know everything the other person is willing to share about themselves!
Debate & Discussion
Debate and discussion both involve a back and forth dialogue between two viewpoints. The goal of a person in a discussion is to convince the other person, or to be convinced themselves if they believe the other person has a better argument. In a discussion, the goal is to get closer to the truth together. In a debate however, the goal is to convince the audience, and the debating parties shouldn't expect to convince the opposing side. Bad things happen when one person thinks they are in a discussion and the other person thinks it's a debate.
Shared reality
This is a less philosophical and more social cousin of the discussion. In a shared reality conversation, the participants' goal is not as much about getting closer to objective truth as it is about establishing a common social truth. It may be that there is not much objective truth in the topic to be had in the first place; it could be a matter of establishing taste or social norms.
Argument
An argument is like discussion and shared reality in that the two parties are looking for a common conclusion. In an argument however, the participants have put aside the principle of charity that says you have to interpret the other person's words in as forgiving way as is possible. Gone is also the idea that there can be a conclusion that will perfectly satisfy both parties: a gain by one party has to at least somewhat come at the expense of the other party. What prevents an argument from escalating to violence is partly third party social norms against violence. But not resorting to violence is also a way to get some kind of shared legitimacy for the conclusion.
Deeper levels
A lot of what people say to each other is not communicated at the level of the words, but is communicated by what is intentionally brought up or left out in a way that has more meaning in context.
Signalling
In all of the above types of conversation, there is room for the person speaking to use what they say to define themselves for other people. A long conference that doesn't ever veer into shared reality or discussion is probably two people taking turns just signalling to each other, without the self-satisfaction from signalling they would both get bored quickly.