These two concepts frequently are confused. Here are definitions:
Skepticism is a critical attitude, orientation or outlook towards a proposition or a thesis. It typically is characterized by doubt about, or at least dubiousness towards, its substantive truth value. As such, it is an epistemological stance, not an ontological one. The proposition or thesis actually might be so, however, we lack the means to discern or evaluate whether this is the case. As a practical matter we may never be able to acquire the means, or we may lack it in principle. Thus, for example, one might be “skeptical” about whether Jesus actually performed the miracles attributed to him in the synoptic gospels. One needs more evidence, possibly provided by textual examination or scientific inquiry, to determine whether this is so. If such evidence is not forthcoming, or never can be produced, then the skeptic will remain doubtful, and is justified in doing so. Skepticism can be taken to the micro level; for example, Hume was skeptical about the process of induction. For him, it simply was one damn thing after another. Much of modern philosophy is about refuting Hume and trying to justify the validity of causal inferences, scientific theories, etc. Psychology is particularly complex in this regard because while it is epistemically objective (one can look at behavior, or outcomes on psychometric tests, or neuronal processes, etc.) it is ontologically subjective (one never will be able to bridge the explanatory gap between them and phenomena such as first-person mental states or “consciousness”).
Nihilism, on the other hand is an attitude, orientation or outlook of indifference towards the proposition or thesis. The nihilist refuses to engage in an epistemological process of examination, discovery or analysis into its truth value. The nihilist is disinterested in its ontological status. The nihilist neither accepts nor rejects its substantive truth or falsity. For that matter, the nihilist is unable to characterize the proposition or thesis as “meaningful” or “meaningless,” to begin with, because that would entail making a substantive evaluation or value judgment. The nihilist just doesn’t care. Thus, for example, the nihilist neither is a believer nor an athiest (the latter requiring one to adopt a stance towards truth value, i.e., affirmatively disbelieving a theological proposition, or asserting it is false). The nihilist is not even agnostic, because that would require one to adopt an epistemological stance (e.g., “it can’t be proven true or false one way or the other”). The most common mistake people make in trying to define nihilism is that they think it is “about” something. Not only is it not about something, it is not even about nothing. The nihilist doesn’t care that he doesn’t care, and so on recursively. For these reasons nihilism is not particularly useful, except possibly as an extreme counterexample.