Externalists hold that there are mental events that do not supervene merely on physical events internal to the agent's body, but supervene on environmental events as well. Also known as anti-individualism.

For some externalists, mental events, namely those in which concepts are involved, can be seen to be partially constituted by the entities that those concepts are concepts of. This intuition gets pumped by the famous Twin Earth thought experiments of Putnam (1975) and Burge (1979), which I assume are familiar enough to not require recounting. Suffice it to mention that the upshot of the experiments is that the Earthling Pete and his molecule for molecule Twin Earth counterpart Twin Pete have divergent mental characteristics solely in virtue of their divergent environments: Pete's 'water' thoughts are about H2O whereas Twin Pete's 'water' thoughts are about XYZ.
Externalism does not apply solely to natural kind concepts like 'water'. Externalist arguments, using Twin Earth style thought experiments, have been advanced for all sorts of concepts. Further, externalist arguments have also been advanced for non-conceptual mental states, so externalism can apply to any contentful mental state. Some have advanced arguments for the externalist individuation of qualia on the grounds that they are representational in character (Dretske 1995; Tye 1995).
It is important to note that Burge, in formulating his externalist arguments, considered them as arguments explicitly against, among other things, functionalism (Burge 1979). However, the functionalism Burge had in mind was a strictly internalist functionalism--he left as an open issue whether a functionalism that quantified over environmental as well as bodily states of cognizers could be viable. In fact, the dominant reaction in the literature to Burge's arguments has been to retain functionalism, but allow that events in one's environment are part of the causal topology that constitutes mental states.
Pete Mandik