Hermeneutics is all about how to properly interpret a text, and was initiated mainly to deal with the Bible, e.g. Augustine wanted to know how to reconcile the Old and New Testaments, and many more recent folks wanted to know how to interpret the Bible so that the horrible parts aren't taken literally: How can the Book speak to us modern folks? It was given its modern form by Schleiermacher (the episode we did about him wasn't on this topic, but our treatment will give you an idea of his approach), who urged us to learn about the historical distance between us and the text (not just the Bible) and systematically purge ourselves of our modern prejudices so that we can enter into the world of the text, to understand the author's meaning as well as possible.
Gadamer's response is that it's not possible to purge ourselves of our prejudices, and while there are of course bad prejudices that make us willfully misunderstand a text and usually dismiss it, the word "prejudice" really just means hypothesis, and of course we need to constantly make hypotheses and hold forth expectations in order to understand a text at all. It's like Plato's puzzle in the Meno: how can look for knowledge unless you already know it well enough to look for it?
The "hermeneutic circle" traditionally meant going back and forth between the sentence you're trying to interpret and the work as a whole. You need to use your expectations of what the work is in order to interpret any one sentence, but then as you interpret the individual parts, your conception of the whole evolves. Gadamer adds to this going back and forth between the text and your own prejudices, which (good or bad) should be as much as possible brought to light, so that engaging with a text means engaging with yourself too: it's actually just a part of knowing thyself, i.e. a part of philosophy. And moreover, you don't even need a text; this strategy applies whenever you're trying to understand what someone is saying, whether through conversation or through a work of art.
We spent most of our time discussing chapter 4 of Gadamer's most famous work, Truth and Method (1960), "Hermeneutic Circle and the Problem of Prejudices" (we read pages 268-273 and then 291-299). Gadamer was a student of Heidegger, and this is where he directly relates what he's doing to Heidegger's language and project: we are temporal beings, and the way we understand something has a "forestructure" (i.e. our expectations).
Gadamer thinks that Heidegger was onto something in describing all of this not as an interpreter consciously experiencing a text, explicitly handling hypotheses about it and using reason to come to the best conclusion. This doesn't capture how unconscious this fore-understanding is, and how permanent our situation of being stuck in our own bed of prejudice is. Heidegger describes all of this as an ontological matter, i.e. as a matter of our modes of being, where we are not fundamentally separate from these things that we interrelate with, not Cartesian subjects looking at objects but fundamentally a unit. Gadamer describes interpretation with the image of a game, where we likewise don't think of a game primarily as individuals sitting back and cogitating at each other, but as an interactive whole governed by its rules: we don't play a game with others so much as the game plays both us and the others. Likewise, when you fully enter into hermeneutics, you're putting yourself on the line, putting your preconceptions at risk, and you don't come out the same person exactly as when you started. In a sense, it's your prejudices, your traditions, that are acting and being subtly altered, through you. It's not like you can choose your traditions, or take them off like clothes. You are just one element in this complex interaction, though a key element, as it's your decision to throw yourself into this hermeneutic game, and your skill that determines how well you let the Other speak to you.
We also discussed the essay "Aesthetics and Hermeneutics" (1964), which criticized Kant's view of appreciation as too consciousness-oriented per the above. If art is just something to be enjoyed, and the only pure art is all about form, then art becomes purely a luxury, an idle passtime of the rich. Gadamer emphasizes that good art is always saying something, even when no words are involved. It is part of a culture, meant to be doing things in people's lives. Moreover, a work's meaning always exceeds what the author explicitly intended; to "mean" anything is to enter a fundamentally public world of expression where idioms are historically developed, so you're really not delving into the mind of the artist in connecting with the work but learning more about the meaning expressed, including how it relates to you in your historical situation that may be different than the artist's. A great work will always have more to say to future generations, so long as they enter into hermeneutics and are open to actually listening to it.
"The Universality of the Hermeneutical Problem" (1966) described (among other things) how this hermeneutic approach relates to science: First, title of his book Truth and Method is misleading, in that the "method" part is all about how hermeneutics can't actually be a method with strict rules laid out, whereas science much more resembles this kind of method. Hermeneutics (and the social sciences in general) shouldn't try to ape the natural sciences, and it's unfortunate that our whole society has become so dominated by the ethos of science that we think anything that doesn't use the kind of reasoning employed by science must be simply bullshit. Gadamer points out that actually, underneath the scientific method, there still is (unscientific!) hermeneutics going on: you have to decide what experiments to perform next, what knowledge is worth seeking, etc. (All this was previously discussed in different terms on our Thomas Kuhn episode.) We need hermeneutics, for example, to tell us how to interpret statistics, and a really great scientist succeeds not just by following a method, but through imagination and consequent intuitive leaps (such as Newton being inspired by the falling apple).
Moreover (and this gets into the territory of Heidegger's "The Question Concerning Technology"), trying to apply the methods of science to everything leaves us blind to these value questions surrounding science, and too apt to look at everything in terms of efficiency, as resources to be used. Scientists ignore the practical uses to which their findings may be put, and we lose the ability to see what is fundamentally questionable in the way things are being done.
Finally, we spent a little time on "Hermeneutics as Practical Philosophy" (1972), which clarified the "non-scientific" character of hermeneutics by bringing in Aristotle and how he contrasts techne (anything involving a method, which could be shoemaking but also technological science) from prohairesis, i.e. praxis that involves human preferences, which is the source of doing ethics and politics and really all philosophy that's not purely a matter of trying to find out facts.
So the distinction is not the modern one between being practical (researching new medicines, building things, getting a job) and intellectual masturbation of one form of another. That picture involves the same sort of mistake as the alienated model of art described above. No, we need to understand that philosophy too (hermeneutics being a major part of this) involves praxis. It is oriented toward human good, and far from being "useless," it provides necessary orientation without which our science and industry are blind machines, chugging away building the needs that nobody needs (to quote The Lorax, not Gadamer). And this "orientation" is not a one-time thing, but requires constant (or at least frequent) engagement, openness, and sympathetic communication, i.e. philosophical hermeneutics as a way of life.
(To clarify, Aristotle recognizes theoretical philosophy too, i.e. pure science and math -- he puts theology in there too -- which is of course not praxis. So the distinction between that kind of science and philosophy is not between knowledge and practice, but within the realm of knowledge. This doesn't undermine his criticism about the way we dismiss philosophy as impractical in our culture today.)
Get the texts:
"The Universality of the Hermeneutical Problem" and "Hermeneutics and Aeshetics" are both in the collection Philosophical Hermeneutics, which also has an excellent, lengthy introduction by David E. Linge. You can read them online; pages 3-17 and 95-104.
"Hermeneutics as Practical Philosophy" is in The Gadamer Reader: A Bouquet of the Later Writings. Read it online (pages 227-245).