So we've established that the central question of literary theory is "how do we understand one another?"
What are theoretical "approaches"? Barthes conceived of them as metalanguages. In other words, an approach is a language, a set of terms, that explains another language, that of literature itself. Now I think many people are tired of approaches or readings that take a theory and apply it to a text, but that is one still prevalent way of thinking about theory, and useful as a kind of shorthand for what we do.
By taking an approach and applying it, you are doing criticism but not yet doing theory. Doing theory is considering the conditions of understanding and the validity of those particular metalanguages. A critique of psychoanalysis, for example, in its hermeneutical limits, would be doing theory, but not a psychoanalytic application to a particular text.
The approach chosen has to do with several questions. Ideological approaches have to do with the question of "who gets to decide what the interpretive framework is?" Where does the authority come from? Then, from there, making judgments about the text from a particular ideological slant.
Hermeneutic approaches have to do with the metaconditions of understanding itself.
Approaches to language or form have to do with the specificity of literary language itself.
Here's the thing: all approaches have to have all these dimensions. There is always someone deciding about power and authority, there is always some determination of what makes understanding possible, and there is always, one hopes, some engagement with what makes literature literary. Of course, some criticism might seem to be ideological strip-mining or formalist wanking, but all these dimensions are present all the time, whether explicitly or implicitly. For example, an ideological critique is always based on some theory of representation and literary language. Is the ideological charge of a text conscious or unconscious; is it presented literally or through allegory? Etc...