Non-reductive materialism is the view that mental states are multiply realizable. As humans, dolphins, Martians and androids might all come under the intentional predicate of "believing that the fish are in the bucket," say, it follows that type-to-type identity fails: the type of intentional state we call "believing that the fish are in the bucket" cannot be identified with any specific type of neural state (some human neural state for example). The "materialism" part of non-reductive materialism is token-to-token identity: every token intentional state is identical, on this view, to some token physical state. This is a basic premise of functionalism: functional descriptions must replace physical descriptions because functions are multiply realizable (or, functions supervene on physical systems).
We can think about the difference between externalist non-reductive materialism and internalist non-reductive materialism. The internalist thinks that intentional predicates pick out states that are in the head (or body anyway, if you wish to be more careful). To say that "He believes that the fish are in the bucket" is to say that there is a particular state of affairs in his head, presumably some neural one. On the internalist view, what is multiply realizable is this internal process that in humans is essentially neural. We can see that this view is closely tied to representational models of nervous system function: if believing that the fish are in the bucket is a state of affairs (or a process) that is happening entirely in the subject's head this may entail that "fish," "the bucket" and so forth are somehow (images? formal symbols?) represented in the head.
But there is also the option of externalist non-reductive materialism. On this view intentional predicates apply to whole, embodied persons interacting with their environment. This looks to me to be right. Brains don't think any more than stomachs eat lunch: stomachs digest, persons eat lunch. Persons think, brains do...what? Seeing this question feels like progress. The externalist view is that mental predicates do not refer to brain states. That insight is interesting on the mental side, but it also washes back onto the question of what it is that neural processes in fact accomplish.