What is a comprehensive definition of Philosophy? Alternatively, is it impossible to define Philosophy? This is a pseudo-meta question, but it seems like it belongs here.
Philosophy is the practice of discovering new ideas and analyzing claims. The two parts can't themselves claim to be "philosophy" on their own since the very name means lover of wisdom. Can the collector of first editions who never reads them be considered a lover of books? What about the woman who reads every book she finds at the library, but never owns one? What about the man who reads romance novels, but will never look at non-fiction? What about the reviewer who reads books for their job, but never for their own pleasure. Or the reader who never finds fault in what they read? No! The lover of books will not despise them in these ways. In the same way, the true philosopher will not tire of seeking the truth.
It's been claimed that the work of philosophy can been distilled into the conflict between Plato and Aristotle. Roughly speaking, the difference between the two can be found in the "discovery" portion of philosophy and thinkers of all sorts may be divided into the Platonic and the Aristotelian camps. If you think from the abstract to the concrete (top-down) and believe that given a sufficient set of axioms you can deduce all truth, then Plato is your man. On the other hand, if you start with observations of the world and use inductive reasoning to discover truth (bottom-up), you are a philosophical child of Aristotle. Both men pioneered new territory in the tools of philosophy (abstract reasoning, logic, preservation of the ideas of previous thinkers, skepticism, and so on), but each remains relevant long after their specific ideas were shown to be wrong. We remember them because of their unique approaches to how we discover truth.
All ages seem to have their controversies which seem to fall on lines drawn between the two Greek philosophers: Anselm of Canterbury v. Gaunilo of Marmoutiers, Bertrand Russell v. Kurt Gödel, René Descartes v. Blaise Pascal, Leonhard Euler and Daniel Bernoulli v. Gottfried Leibniz and Christian Wolff, Bishop Berkeley v. Samuel Johnson, Immanuel Kant v. David Hume, etc. That's not to say philosophy hasn't moved on, but rather each bit of new ground has features that resemble the philosophical territory covered by those great Athenians from the 4th-century BC.
Much of the thinking that once fell under the banner of Philosophy has been pushed into more specialized disciplines. Practically every department at universities can be traced back to philosophy at some point in the distant past. Even the study of music (studied by Pythagoras and his followers), poetry (Aristotle wrote the book on it), political science (Plato's Republic is standard reading in both departments), and law (the Socratic method plays an important part in many law schools) can be seen as sub-disciplines of philosophy. If Mathematics is "The Queen of the Sciences"1, then Philosophy is her King.
1) It was too much a stretch to use the original Queen: theology. Thankfully Carl Friedrich Gauss reappropriated the term.
I think Wittgenstein is very illuminating in this respect. Some relevant quotations, dealing with some aspects of the question (PI stands for Philosophical Investigations):
PI§109 [...] Philosophy is a battle against the bewitchment of our intelligence by means of our language.
PI§126 [...] One might also give the name "philosophy" to what is possible before all new discoveries and inventions.
PI§127 The work of the philosopher consists in assembling reminders for a particular purpose.
And as far as philosophy speaking of itself posing a problem:
PI§121 One might think: if philosophy speaks of the use of the word "philosophy" there must be a second-order philosophy. But it is not so: it is, rather, like the case of orthography, which deals with the word "orthography" among others without being second-order
In general I highly recommend anyone troubled by the question 'What is philosophy?' to read PI§89-138 - one of the most brilliant self-reflections on the purpose and nature of philosophy in the history of thought. Of course you have to be minimally aware of the context of Wittgenstein's thought, but I feel these passages are elucidatory nonetheless.
Here are some popular definitions of philosophy.
The word "philosophy" comes from the Greek φιλοσοφία (philosophia), which literally means "love of wisdom"
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. It is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational argument.
Philosophy is the love of, or the search for, wisdom or knowledge. It's theory or logical analysis of the principles underlying conduct, thought, knowledge, and the nature of the universe"
Another dictionary says,
Philosophy is the most basic beliefs, concepts, and attitudes of an individual or group
I think the study of philosophy can get accurately said to seek for the most comprehensive understanding of things. Epistemology seeks the most comprehensive understanding of knowledge. Metaphysics seeks for the most comprehensive understanding of reality. Ontology seeks for the most comprehensive understanding of being or existence. Ethics seeks for the most comprehensive understanding of proper conduct. Aesthetics seeks for the most comprehensive understanding of art. Logic seeks for the most comprehensive understanding of reasoning (hence the plurality of logics, and symbolic logic as a formal discipline). And the philosophy of particular disciplines like the philosophy of biology, physics, etc. seeks for the most comprehensive understanding of those disciplines.
Philosophy is just a word, like many others. As a 'science' it has no scope, and it's entirely based on the difficulties we have in understanding our own language. Look around, there are no genuine philosophical problems. We define words, like "soul" or "God" and then, we talk endlessly about them, that's Philosophy.