The term Continental philosophy has no singularly accepted formal definition, nor does it even signify a “you know it when you see it” kind of activity, because it is not really a distinguishable activity at all. Indeed, most people who study philosophy on the continent have no idea that it is “continental philosophy” they are studying, but simply see it as philosophy proper. The term continental philosophy itself was not used by any academic departments until the 1970s and it hadn't been until the 1920s that really any kind of major divide in philosophy became noticeable at all.
This is because continental philosophy is, by default so to speak, defined no more explicitly than as that which isn't analytic philosophy. Analytic philosophy emerged in the early twentieth century as an explicit rejection of the history of philosophy as something littered with logical mistakes and conceptual unclarity. The founding belief of the analytic school was that if you were just as precise as possible about the language you were using, most philosophical problems would either disappear, become meaningless or solve themselves. Whilst this position is not explicitly part of analytic philosophy today, it still structures its practices and its prejudices. It is still very common to find a suspicion in the mind of the analytic philosopher that the history of philosophy is just a series of mistakes largely due to imprecision.
The reduction of philosophy to logic, that is from the study/theory of concepts to the study/theory of the formal structure by which concepts are linked, has been partly an obscuring force in philosophy. It has led some to believe that the formal structure is the philosophy itself, whereas it is merely an insubstantial, if more precise way, of formulating ideas. It is, however, ideas that are meat on the bone.
The transition point at the founding of analytic philosophy was immediately preceded by centuries of philosophical conversation--Aristotle speaking to Kant, and Hegel speaking to Descartes--with all these voices, through generations of philosophers and their followers, going back to each other’s works and responding to them. It is this broad, far reaching conversation that continues in continental philosophy today and which partly explains why it is much harder to jump into it than analytic philosophy often is. To do analytic philosophy, one simply has to learn logical analysis and address isolated problems, whereas to take part in the “continental conversation” you not only have to know what the philosophers themselves said but how they have been interpreted, reinterpreted and replied-to throughout time.
Continental philosophy is often treated or defined by its relationship to Kant. At the fracturing point in philosophy at the beginning of the twentieth century, the philosophical conversation was continuous with Kant – of course! And it is the vaguely Kantian strains of thought that analytic philosophers seemed mostly to take issue with. However, I think this neat conceptual divide (between people who take Kant seriously and people who do not) does not explain why their practices, approaches and styles of philosophical methodology differ. It seems to explain much more to say that analytic philosophers abandoned philosophy as it was at the beginning of the 20th century and re-founded it on the logical analysis of language, whereas continental philosophers were those who didn’t do this but simply continued on with the philosophical conversation.” And this is why Kant is still around: he was never explicitly forgotten.