Continental rationalism is an approach to philosophy based on the thesis that human reason can in principle be the source of all knowledge. It originated with René Descartes and spread during the 17th and 18th centuries, primarily in continental Europe. In contrast, the approach known as British Empiricism held that all ideas come to us through experience, either though the five external senses or through such inner sensations as pain and pleasure, and thus that knowledge (with the possible exception of mathematics) is essentially empirical. At issue is the fundamental source of human knowledge, and the proper techniques for verifying what we think we know (see Epistemology).
The distinction between rationalists and empiricists was drawn at a later period, and would not have been recognised by the philosophers involved. Also, the distinction wasn't as clear-cut as is sometimes suggested; for example, the three main rationalists were all committed to the importance of empirical science, and in many respects the empiricists were closer to Descartes in their methods and metaphysical theories than were Spinoza and Leibniz. Thus, although it can be useful when organising courses or writing books, the distinction is less useful philosophically.
Rationalists typically argued that, starting with intuitively-understood basic principles, like the axioms of geometry, one could deductively derive the rest of knowledge. The philosophers who held this view most clearly were Baruch Spinoza and Gottfried Leibniz, whose attempts to grapple with the epistemological and metaphysical problems raised by Descartes led to a development of the fundamental approach of rationalism. Both Spinoza and Leibniz thought that, in principle, all knowledge – including scientific knowledge – could be gained through the use of reason alone, though they both accepted that in practice this wasn't possible for human beings except in specific areas such as maths.
Descartes, on the other hand, was closer to Plato, thinking that only knowledge of eternal truths – including the truths of mathematics, and the epistemological and metaphysical foundations of the sciences – could be attained by reason alone; other knowledge required experience of the world, aided by the scientific method. It would perhaps be most accurate to say that he was a rationalist with regard to metaphysics, but somewhat akin to the empiricists with regard to the physical sciences.
Immanuel Kant started as a traditional rationalist, but after studying David Hume's works which "awoke [him] from [his] dogmatic slumbers", he developed a distinctive and very influential rationalism of his own which attempted to synthesise the traditional rationalist and empiricist traditions.
The more modern usage of the term "rationalist" refers to the belief that human behaviour and beliefs should be based on reason — a belief shared by continental rationalists and empiricists alike (see Rationalism).