Noncognitivism, when pursued most consistently, means refusing to engage in debates over the truth of religious claims unless and until religious apologists can provide coherent definitions, claims, and ideas to debate. So long as religious claims are meaningless, there is nothing to even discuss, much less debate.
Defending Theological Noncognitivism
Proponents of theological noncognitivism offer a variety of arguments, depending on what they think about meaning.
Verificationists argue that religious language is not meaningful because its empirical claims cannot be verified, even in theory. Attributists argue that there are no positive attributes that can be ascribed to entities like "god," and entities without attributes are meaningless.
Others argue that many basic religious claims do not express real propositions or the propositions they try to express are incoherent. Just because a proposition is grammatical does not mean that it is meaningful or coherent; for example, "colorless green ideas sleep furiously" is not meaningful even though it is grammatical and understandable. Theological noncognitivism argues that religious language belongs in the same category.
Theological Noncognitivism & Atheism
Noncognitivism is compatible with some but not necessarily all forms of atheism. Noncognitivism supports at least weak atheism because if one does not think that religious language about gods is meaningful, then they aren't going to believe in any gods. It's implausible that noncognitivism would ever be compatible with theism.
It's debated whether noncognitivism is compatible with strong atheism. If noncognitivism is right that religious claims about gods cannot be true or false, then it's not reasonable to insist that it's false that any gods exist, which is strong atheism. It's reasonable to not accept as true an incoherent claim, but it's probably not reasonable to assert as false an incoherent claim.