You may often hear philosophers or theologians categorized as being either continental or analytical. For instance, continental philosophers/theologians include such names as Friedrich Nietzsche, Soren Kierkegaard, Martin Buber, Jacques Derrida, Franz Rosenzweig, Jürgen Moltmann, Rowan Williams, Peter Rollins, Slavoj Žižek, Edmund Husserl, Jürgen Habermas, and Alain Badiou. On the analytic side of the spectrum you will find such names as Alvin Plantinga, Nicholas Wolterstorff, Noam Chomsky, William Lane Craig, Daniel Dennett, and Bertrand Russell.
Continental philosophy refers generally to the philosophy done on mainland Europe from Kant onwards. As a philosophical approach it is primarily focused on tackling problems through the dialogue between major philosophical figures (e.g. Kant, Hegel, Deleuze, Sartre, Heidegger, Foucault, Levinas, Derrida, etc), and the meaning and socio-historical context of their thought and oeuvres. Continental philosophy can be thought of as the “history of philosophy”, where knowing the history of philosophy is a key part to the doing of philosophy, and so in reading the writings of a continental philosopher, one will see they are concerned with things such as the history, process, and narrative of philosophical thought. The continental style of writing is typically seen as being much more literary with the frequent use of technical language, possibly leading to less emphasis on clarity.
Analytic philosophy is generally thought to stem from Gottlob Frege, and is quite prevalent in England, America, and Australia. It is more focused on the philosophical problem in and of itself, seeing the history of thought on it as irrelevant. It is immersed in formal, mathematical logic, and even in an area where the use for such logic is lacking (e.g. ethics), attention is then laid upon the method of valid deductive inference. Analytic philosophy is typically seen as a science, or at least as being continuous with the natural sciences (whereas I have seen continental philosophy being classified as almost antithetical to the natural sciences and instead subsumed under the social sciences).
In short, the distinction between continental and analytic philosophy in practice is that the former centers on concepts in their contingent historical context (as put forth by a specific philosopher), and the latter centers on issues outside of any historical context (it extracts and isolates a concept or problem from its historical context); a continental philosopher might say something like, “I am studying Hegel”, whereas the analytic philosopher would say something more like, “I am studying the content of perception.” Or to convey the term in more technical language (as any good wannabe continental philosopher would do), one can say that the continental approach deals with the meta-ontological, whereas the analytical approach deals with the ontic 0f the ontological.
Personally I prefer the continental approach (just like my breakfast). Why? Because I don’t think that a philosophical concept or problem is an ideal, timeless entity that came to us in a vacuum or from some realm of unadulterated rational thought. Thus I think it is beneficial to approach philosophical problems or concepts through the continental method. Though, as I alluded to before, continental philosophy can seem purposefully obscurantist and quite convoluted at times, which can all too easily lead to one throwing a book across the room while screaming, “what the hell are you talking about Žižek!” But this should not necessarily be seen as being reflective upon any inherent pitfall of the continental philosophical tradition, but is rather borne out of a mind that is used to a more analytical approach.
Finally, I would be remiss if I didn’t make the disclaimer that some do not think there actually is a meaningful distinction between analytic and continental philosophy, but that it is more of an unnecessary artificial human construction, in which case you can just ignore everything I said!