20th Century philosophy has been dominated to a great extent by the rivalry between two very general philosophical traditions, Analytic Philosophy (the largely anglophone mindset that philosophy should apply logical techniques and be consistent with modern science) and Continental Philosophy (pretty much a label for everything else. As the name says, these thinkers are mainly based in mainland Europe. In very general terms, Continental Philosophy consists of the European tradition, it tends towards historicism, and rejects scientism and positivism. It does not want to reduce philosophy to a theory of science, or language.
Analytic Philosophy
An important precursor of the Analytic Philosophy tradition was the Logicism developed during the late 19th Century by Gottlob Frege. Logicism sought to show that some, or even all, of mathematics was reducible to Logic, and Frege's work revolutionized modern mathematical Logic. In the early 20th Century, the British logicians Bertrand Russell and Alfred North Whitehead continued to champion his ideas (even after Russell had pointed out a paradox exposing an inconsistency in Frege's work, which caused him, Frege, to abandon his own theory). Russell and Whitehead's monumental and ground-breaking book, "Principia Mathematica" was a particularly important milestone. Their work, in turn, though, fell prey to Kurt Gödel's infamous Incompleteness Theorems of 1931, which mathematically proved the inherent limitations of all but the most trivial formal systems.
Both Russell and Whitehead went on to develop other philosophies. Russell's work was mainly in the area of Philosophy of Language, including his theory of Logical Atomism and his contributions to Ordinary Language Philosophy. Whitehead developed a metaphysical approach known as Process Philosophy, which posited ever-changing subjective forms to complement Plato's eternal forms. Their Logicism, though, along with Comte's Positivism, was a great influence on the development of the important 20th Century movement of Logical Positivism.
The Logical Positivists campaigned for a systematic reduction of all human knowledge down to logical and scientific foundations, and claimed that a statement can be meaningful only if it is either purely formal (essentially, mathematics and logic) or capable of empirical verification. The school grew from the discussions of the so-called "Vienna Circle" in the early 20th Century (including Mauritz Schlick, Otto Neurath, Hans Hahn and Rudolf Carnap). In the 1930s, A. J. Ayer was largely responsible for the spread of Logical Positivism to Britain, even as its influence was already waning in Europe.
The "Tractatus" of the young Ludwig Wittgenstein, published in 1921, was a text of great importance for Logical Positivism. Wittgenstein is one of the 20th Century's most important philosophers. A central part of the philosophy of the "Tractatus" was the picture theory of meaning, which asserted that thoughts, as expressed in language, "picture" the facts of the world, and that the structure of language is also determined by the structure of reality. At the time he was convinced that the publication of the "Tractatus" had solved all the problems of philosophy: They exist only because language is used in the wrong way. Wittgenstein later abandoned his early work. He began to see language not as a model of reality, but as a system that can generate meaning intrinsically. He looked at language as a kind of game which generates its own rules and which we use in different ways. This approach led to the development of "Ordinary Language Philosophy."
Ordinary Language Philosophy shifted the emphasis from the ideal or formal language of Logical Positivism to everyday language and its actual use, and it saw traditional philosophical problems as rooted in misunderstandings caused by the sloppy use of words in a language. Some have seen Ordinary Language Philosophy as a complete break with, or reaction against, Analytic Philosophy, while others have seen it as just an extension or another stage of it. Either way, it became a dominant philosophic school between the 1930s and 1970s, under the guidance of philosophers such as W. V. O. Quine, Gilbert Ryle, Donald Davidson, etc.
Quine's work stressed the difficulty of providing a sound empirical basis where language, convention, meaning, etc, are concerned, and also broadened the principle of Semantic Holism to the extreme position that a sentence (or even an individual word) has meaning only in the context of a whole language. Ryle is perhaps best known for his dismissal of Descartes' body-mind Dualism as the "ghost in the machine", but he also developed the theory of Philosophical Behaviorism (the view that descriptions of human behavior need never refer to anything but the physical operations of human bodies) which became the standard view among Ordinary Language philosophers for several decades.
Another important philosopher in the Analytic Philosophy of the early 20th century was G. E. Moore, a contemporary of Russell at Cambridge University (then the most important center of philosophy in the world). His 1903 "Principia Ethica" has become one of the standard texts of modern Ethics and Meta-Ethics, and inspired the movement away from Ethical Naturalism (the belief that there exist moral properties, which we can know empirically, and that can be reduced to entirely non-ethical or natural properties, such as needs, wants or pleasures) and towards Ethical Non-Naturalism (the belief that there are no such moral properties). He pointed out that the term "good", for instance, is in fact indefinable because it lacks natural properties in the way that the terms "blue", or "smooth," have them. He also defended what he called "common sense" realism (as opposed to Idealism or Skepticism) on the grounds that common sense claims about our knowledge of the world are just as plausible as those other metaphysical premises.
Continental Philosophy
On the Continental Philosophy side, an important figure in the early 20th Century was the German Edmund Husserl, who founded the influential movement of Phenomenology. He developed the idea, parts of which date back to Descartes and even Plato, that what we call reality really consists of objects and events ("phenomena") as they are perceived or understood in the human consciousness, and not of anything independent of human consciousness (which may or may nor exist). Thus, we can "bracket" (or, effectively, ignore) sensory data, and deal only with the "intentional content" (the mind's built-in mental description of external reality), which allows us to perceive aspects of the real world outside.
It was another German, Martin Heidegger (once a student of Husserl), who was mainly responsible for the decline of Phenomenology. In his groundbreaking "Being and Time" of 1927, Heidegger gave concrete examples of how Husserl's view (of man as a subject confronted by, and reacting to, objects) broke down in certain (quite common) circumstances, and how the existence of objects only has any real significance and meaning within a whole social context (what Heidegger called "being in the world"). He further argued that existence was inextricably linked with time, and that being is really just an ongoing process of becoming (contrary to the Aristotelian idea of a fixed essence). This line of thinking led him to speculate that we can only avoid what he called "inauthentic" lives (and the anxiety which inevitably goes with such lives) by accepting how things are in the real world, and responding to situations in an individualistic way (for which he is considered by many a founder of Existentialism). In his later work, Heidegger went so far as to assert that we have essentially come to the end of philosophy, having tried out and discarded all the possible permutations of philosophical thought (a kind of Nihilism).
Existentialism
The main figure of the existentialist movement was Jean-Paul Sartre (along with his French contemporaries Albert Camus, Simone de Beauvoir and Maurice Merleau-Ponty). A confirmed Atheist and a committed Marxist and Communist for most of his life, Sartre adapted and extended the work of Kierkegaard, Nietzsche, Husserl and Heidegger, and concluded that "existence is prior to essence" (in the sense that we are thrust into an unfeeling, godless universe against out will, and that we must must then establish meaning for our lives by what we do and how we act). He believed that we always have choices (and therefore freedom) and that, while this freedom is empowering, it also brings with it moral responsibility and an existential dread (or "angst"). According to Sartre, genuine human dignity can only be achieved by our active acceptance of this angst and despair.
Structuralism, Post-Structuralism, Deconstruction
In the second half of the 20th Century, three main schools (in addition to Existentialism) dominated Continental Philosophy. Structuralism is the broad belief that all human activity and its products (even perception and thought itself) are constructed and not natural, and that everything has meaning only through the language system in which we operate.
Post-Structuralism is a reaction to Structuralism, which stresses the culture and society of the reader over that of the author). Post-Modernism is an even less well-defined field, marked by a kind of "pick'n'mix" openness to a variety of different meanings and authorities from unexpected places, as well as a willingness to borrow unashamedly from previous movements or traditions.
The radical French historian and philosopher Michel Foucault has been associated with all of these movements (although he himself always rejected such labels). Much of his work is language-based and, among other things, he has looked at how certain underlying conditions of truth have constituted what was acceptable at different times in history, and how the body and sexuality are cultural constructs rather than natural phenomena. Although sometimes criticized for his lax standards of scholarship, Foucault's ideas are nevertheless frequently cited in a wide variety of different disciplines.
Mention should also be made of Deconstructionism (often called just Deconstruction), a theory of literary criticism that questions traditional assumptions about certainty, identity and truth, and looks for the underlying assumptions (both unspoken and implicit), as well as the ideas and frameworks, that form the basis for thought and belief. The method was developed by Jacques Derrida. His work is cerebral and self-consciously "difficult", and he has been repeatedly criticized for his pseudo-philosophical style.