Short Answer
Emotivism is a meta-ethical theory that says that ethical sentences are not truth-bearing propositions, but propositional attitudes expressing emotion. Intuitionism, on the other hand, is a class of ethical systems that say that some ethical truths can be known with without inference, and since intuitionism believes the opposite of emotivism, that moral truths can be bearers of truth, it is a type of theory that belongs to cognitivism.
Long Answer
Languages and Inferences
Okay, the question you have has some complexity to it that requires a familiarity with some basic philosophical terminology, so let's flesh that out first so we are on the same page:
- sentence - An utterance, an expression, a formula, a string, or an instance of syntax. 'The snow is white' is an example of a sentence.
- proposition - The meaning ascribed to a sentence, confusingly expressed in written communication as a sentence. So, 'The snow is white' is a true proposition if the snow is actually white. 'The snow is green' is a false sentence, because snow is white.
- syntax-semantics dichotomy - From above, we can see that sentences try to express propositions or languages try to express ideas. Misunderstanding happens when sentences do not adequately communicate propositions, or language doesn't adequately express ideas.
- inference - Now, when we bump propositions against each other, other propositions come out of our heads. For instance, 'Socrates is in the kitchen.', 'The kitchen is in the house.', therefore we know 'Socrates is in the house.'. See how that works?
Now, the bread and butter of philosophers is trying to make sense of sense and reference, an idea that was fleshed out very well until Gottlob Frege at the end of the 19th century despite thousands of years of philosophy. What's important to answer your question is that we three things going on. We have people uttering ethical statements (sentences), people meaning ethical ideas (meanings), and people deciding whether ethical statements and ideas are consistently true (proof and consistency). Philosophers distinguish between sentences and propositions (sometimes) by use-mention distinction, and (sometimes) use delimiters. In the sentence, "'A' is A", 'A' is the statement, and A is the meaning. Reread that because it's implicit in language use and can be confusing, particularly because it's relative to context (The sentence "'A' is A" an example of a third layer of language) and easily confusing.
Ethical and Meta-Ethical Systems
'Killing is wrong' (K) says ethicist. 'Abortion is wrong' (A) says ethicist. 'Death penalty is right' (D) says ethicist. We can see that there might be some problems. Like? Well, K seems to subsume A, so K and A are consistent. Yet, K seems to reject D, so they seem inconsistent. Now, we're not interested in the argument, but how arguments are evaluated, so let's take a look at how this fits with your question.
In cognitivism, we accept that K, A, and D have truth values. Using simple logics, we can say that either K or ~K that is: killing is either right or wrong. But we presume that one of these is true. Now, how do we know? Some ethicists argue we have to have justification that K or ~K is the case. We simply don't pick a value at random. If that's the case, then you do not KNOW K or ~K unless you can PROVE K or ~K. What constitutes proof? That depends on your theory of justification. To simplify, most philosophical proof is predicated on the notion of justified, true belief (JTB), thought a clever man named Gettier has caused some heartache with this theory.
But, intuitionalism rejects the idea that to KNOW a truth is to PROVE a truth with something like JTB. An intuitionalist says that K is true simply because we have a moral sense as humans, that it is intuitively true that killing is just wrong. Now, to some philosophers, this is just unacceptable. How can you just KNOW things without arguments to justify them? In any case, intuitionalism believes that moral reasoning starts partially with moral intuition even if they are rooted in emotion, nice and simple, right? Moral claims are partially derived from logical intuitions that are driven by emotions. It is the fact that moral claims are logical and have truth that makes this a cognitivist theory.
Emotivism is a whole 'nother animal of being. It says, that K is not a truthful claim, because it is neither true nor false. It is simply a proposition that says 'I don't like killing because of the way it makes me feel'; and to talk about truths about this is meaningless. A.J. Ayer was a logical positivist, and to the logical positivists ethical claims simply don't have much value, as the logical positivists tried to eliminate metaphysics from science and logic. (Spoiler alert, they failed.) To an emotivist, feelings cause ethical claims that look logical, but are empty of logic much in the way saying I'm 56% certain isn't really related to the idea of 56/100. It's an example of false precision.
So, emotivism simply claims that ethical systems are conversations about feelings and don't have anything to do with truth at all, largely because they are subjective expressions that have no truth since truth is an objective thing; cognitivism disagrees. Intuitionism, a type of cognitivism, says ethical discussions ARE about truth, and that it's okay that some of the truths start with our intuitions. Emotivism and cognitivism are meta-ethical theories because they have propositions that they claim to be true about ethical theories.
