The paradigmatic questions of meta-philosophy are questions concerning the nature of philosophical problems (or, as some would prefer, puzzles), and the proper methods for setting about to answer them. Another set of related questions, particularly important to ancient philosophers, dealt with the value of philosophy and philosophical knowledge, and the closely related problem of the relationship between philosophical criticism and ordinary life. Other questions, which are perhaps of somewhat more concern to scientifically-minded modern philosophers, are concerned with the relationship between philosophy and the natural sciences, and the question of whether philosophy can make progress in the same sense as the special sciences.
How is meta-philosophy possible?
Philosophers are notorious for being able to make the most commonplace topic collapse into a self-referential hall of mirrors, and the existence of "meta-philosophy" may seem the clearest confirmation imaginable. It may seem that the whole affair cannot help but fall into a massive exercise in circular reasoning; after all, what method could you use to philosophize about philosophical method? If you don't have a method, what can you use to find one? And if you do have one, why go looking? It seems that meta-philosophical questions can hardly be asked unless they are already answered.
Nevertheless, while it certainly seems weird for philosophy to turn its own methods upon itself, it's far from clear what other methods might be appropriate for addressing such questions. After all, if philosophy of language is a valid study, then it seems that one of the things that it can validly study is the nature of philosophical language, and if epistemology (the theory of knowledge) is a valid study, then it seems that one of the things it can validly study is philosophical knowledge. The lesson here, then, may not be so much that philosophy cannot address questions of philosophical method, but rather that questions of the methods and nature of philosophy must be (as it were) explored from the inside, rather than discovered from some impossible outside perspective. A philosopher who does meta-philosophy, then, is not taking some special standpoint from outside of philosophy to look in and determine what philosophy is; rather, she is doing exactly what she does in all other sorts of philosophy. If this is the case, then there is a certain flattening of meta-philosophy back into the mainstream of philosophical inquiry. As Ludwig Wittgenstein was fond pointing out, by way of analogy, when you learn spelling, one of the words you learn to spell is "SPELLING." But we don't speak of "spelling of the second-order."
It may not be surprising, then, to find that many seminal works of philosophy end up in meta-philosophical reflection as a central part of doing philosophy in what seem to be entirely different fields. Although Ludwig Wittgenstein's Tractatus Logico-Philosophicus and Philosophical Investigations are works directly concerning logic, the philosophy of language, and the philosophy of mind, the nature of philosophical puzzles and philosophical understanding is central to all of the discussions; Wittgenstein frequently diagnoses philosophical errors as involving confusions about the nature of philosophical inquiry. Nor is this a new feature in the philosophical world: the tradition goes back to no less a philosopher than Plato, whose dialogues are directly concerned with ethics, but constantly raise questions of philosophical method (most explicitly addressed in the Meno), the value and proper aims of philosophy (in the Apology, Gorgias, Protagoras, etc.), and the proper relationship between philosophical criticism and everyday life (a theme running throughout all his works, but explored most famously in the Republic).
See also: progress (philosophy), philosophy of mathematics, core ontology