There are a number of positions that advocate for the existence of objective moral facts. This is typically called 'moral realism', and is a conjunction of a number of claims:
- Moral claims are truth-apt.
- Moral claims are objective.
- There are at least some true moral claims.
- We can know what these true moral claims are.
We'll focus on 1 and 2. 1 amounts to the claim that non-cognitivism is false: whatever moral claims are, they aren't merely expressions of emotion. Rather, moral claims are on a par with scientific or mathematical claims insofar as they purport to describe some feature of the world - specifically, the moral features. Moral claims are true when they accurately represent the moral facts, and they are false when they don't. The important bit is that they can be true or false at all. So, for instance, the claim that 'sufferring is bad' is not an expression of my or anyone else's emotions or feelings - it is an expression of a fact.
On to 2. Something is true objectively if it is true independent of anyone's thoughts, feelings, beliefs or ideas about that thing. Something is true subjectively if it is not true objectively, i.e. if its truth depends on what people think, feel, believe etc. about it. So 2 says that the truth-values of moral claims do not depend on what anyone thinks, or feels, or believes about them. So, for instance, if it is true that suffering is bad, then it is not because people believe that suffering is bad, or think that it is bad, or feel that it is bad. (More carefully: is that it is not merely because they think, feel, or believe that suffering is bad that it is bad).
3 entails the falsity of error theories, and 4 ensures that moral agents aren't isolated from the moral facts - they're not outside of the domain of things that we can know at all.
So, that is the core of moral realism. It can be fleshed out in other ways as well. If moral realism is true we might ask what kind of facts are moral facts. Are they natural facts, or non-natural facts?
Now, what cogent arguments there are for moral realism will depend on the kind of moral realism you accept. If you're a moral non-naturalist, then Moore and Huemer give compelling cases for their view. Their arguments will typically take the following form: (i) they'll offer up a prima facie plausible example of a moral fact, for instance, 'torturing innocent babies just for fun is wrong'. (ii) they'll claim that any argument which implies the falsity of 'torturing innocent babies just for fun is wrong' will be less prima facie plausible than the truth of the fact that torturing babies just for fun is wrong. (iii) from (i) and (ii) they'll argue that moral realism is correct.
Different arguments exist for the different flavours of realism (see below). The point, however, is that there are cogent arguments for moral realism, and they do deserve to be taken seriously.
Here are some links that can get you started:
http://www.owl232.net/5.htm (this is a chapter from Huemer's book)