A primitive is a basic building block of a system. Complex systems can be decomposed into simpler things, but primitives -- by definition -- cannot.
To provide an example that gives a nice intuition about what a primitive is, consider teaching a child the meanings of different words. If a child asks us "What does `bachelor' mean?", we might break "bachelor" down into other meanings ("`Bachelor' means that someone is a `man' who is `not married'"). However, if a child asks us "What does `red' mean?", we are not likely to do this, because it is difficult to decompose such a basic term. Instead, we are more likely to point to different things that are `red'. In this sense, `red' represents something that we might call a semantic primitive (a basic meaning), while `bachelor' does not.
Primitives are important in cognitive science because of its tendency to view information processors functionally instead of physically. Because of this view, researchers use a methodology called functional analysis to decompose a complex information processor into simpler, functional components. However, if this decomposition is not stopped, the functional analysis goes on indefinitely and falls prey to Ryle's Regress. This means that the functional analysis is not explanatory. Researchers try to escape Ryle's regress by identifying a set of primitive functions which cannot be further decomposed. This set of functions is the functional architecture for cognition.