Recently these terms have become part of the standard jargon of philosophical discourse, and have become central to a couple of important debates.
In contemporary moral philosophy Internalism is the view that moral beliefs function as a motivating factor. That is the internalist believes that there is an internal connection between her belief that "X ought to be done" and her motivation to do X. In the same way an externalist would argue that there is no essential internal connection between moral beliefs and motives. An externalist would say that there's no essential reason that the belief "X is wrong" leads to a desire not to do X. It is likely that this use of these terms comes from W.D. Falk's paper Ought and motivation (1948).
In contemporary epistemology the belief that everything necessary to provide justification for a belief is immediately available to the consciousness is called internalism. Externalism in this context is the view that there are factors other than those which are internal to the believer which govern whether one can be warranted in calling that belief knowledge. One strand of externalism is loosely called the causal theory of knowledge, and reliabilism is sometimes considered to be another strand.
Philosophy of Mind:
Within the context of the philosophy of mind externalism is the theory that mental states are dependent on their relationship to the external world. This is in contrast to both a Cartesian dualism which posits two separable realms for thought and extension (or to use more modern terms, mind and matter), as well as George Berkeley's radical idealism. Externalists generally claim that thoughts or other mental states are caused by external forces, but also that thought, feelings, and all of the other mental states could not exist if except as part of a larger world external to thought. Internalism in this context is precisely the opposite. They would claim that there is nothing essential to the concept of a mental state which requires the existence of an external world.