Internalism and externalism are two opposing ways of explaining various subjects in several areas of philosophy. These include human motivation, knowledge, justification, meaning, and truth. The distinction arises in many areas of debate with similar but distinct meanings. Usually 'internalism' refers to the belief that an explanation can be given of the given subject by pointing to things which are internal to the person or their mind which is considering them. Conversely, externalism holds that it is things about the world which motivate us, justify our beliefs, determine meaning, etc.
In contemporary moral philosophy, motivational internalism (or moral internalism) is the view that moral convictions (which are not necessarily beliefs, e.g. feelings of moral approval or disapproval) are intrinsically motivating. That is, the motivational internalist believes that there is an internal, necessary connection between one's conviction that X ought to be done and one's motivation to do X. Conversely, the motivational externalist (or moral externalist) claims that there is no necessary internal connection between moral convictions and moral motives. That is, there is no necessary connection between the conviction that X is wrong and the motivational drive not to do X. (The use of these terms has roots in W.D. Falk's (1947) paper "Ought" and Motivation).
These views in moral psychology have various implications. In particular, if motivational internalism is true, then an amoralist is unintelligible (and metaphysically impossible). An amoralist is not simply someone who is immoral, rather it is someone who knows what the moral things to do are, yet is not motivated to do them. Such an agent is unintelligible to the motivational internalist, because moral judgments about the right thing to do have built into them corresponding motivations to do those things that are judged by the agent to be the moral things to do. On the other hand, an amoralist is entirely intelligible to the motivational externalist, because the motivational externalist thinks that moral judgments about the right thing to do not necessitate some motivation to do those things that are judged to be the right thing to do; rather, an independent desire—such as the desire to do the right thing—is required (Brink, 2003),(Rosati, 2006).
There is also a distinction in ethics and action theory, largely made popular by Bernard Williams (1979, reprinted in 1981), concerning internal and external reasons for action. An internal reason is, roughly, something that one has in light of one's own "subjective motivational set"---one's own commitments, desires (or wants), goals, etc. On the other hand, an external reason is something that one has independent of one's subjective motivational set. For example, suppose that Sally is going to drink a glass of poison, because she wants to commit suicide and believes that she can do so by drinking the poison. Sally has an internal reason to drink the poison, because she wants to commit suicide. However, one might say that she has an external reason not to drink the poison because, even though she wants to die, one ought not kill oneself no matter what—regardless of whether one wants to die.