Popular historical advocates of some version of the moral sense theory or sentimentalism include the 3rd Earl of Shaftesbury (1671-1713), Francis Hutcheson (1694-1746), David Hume (1711-1776), and Adam Smith (1723-1790). Some contemporary advocates include Michael Slote, Justin D'Arms, Daniel Jacobson, Jesse Prinz, and perhaps John McDowell. Simon Blackburn and Allan Gibbard endorse a non-cognitivist form of sentimentalism.
Generally speaking, (rationalist) ethical intuitionism models the acquisition of such non-inferential moral knowledge on a priori, non-empirical knowledge, such as knowledge of mathematical truths; whereas moral sense theory models the acquisition of such non-inferential moral knowledge on empirical knowledge, such as knowledge of the colors of objects.
Subsequently, Francis Hutcheson (1694-1746) developed a version of moral sense theory. The chief statements of his theory occur in An Inquiry Concerning the Original of Our Ideas of Virtue or Moral Good (1725; Treatise II of An Inquiry Into the Original of Our Ideas of Beauty and Virtue) and An Essay On the Nature and Conduct of the Passions and Affections, With Illustrations Upon the Moral Sense (1728).
Arguably the most prominent defender of moral sense theory in the history of philosophy is David Hume (1711-1776). While he discusses morality in Book 3 of his Treatise of Human Nature (1739-40), Hume's most mature, positive account of the moral sense is found in An Enquiry Concerning the Principles of Morals (1751).
Adam Smith also advanced a form of moral sense theory in his The Theory of Moral Sentiments (1759). Smith focused less on a single faculty of the moral sense and more on the various sentiments that make up the moral feelings that ground moral judgments.
One way to understand the moral sense is to draw an analogy between it and other kinds of senses. Beauty is something we see in some faces, artworks and landscapes. We can also hear it in some pieces of music. We clearly do not need an independent aesthetic sense faculty to perceive beauty in the world. Our ordinary five senses are quite enough to observe it, though merely observing something beautiful is not by itself enough to appreciate its beauty. Suppose we give a name to this ability to appreciate the beauty in things we see: let’s call it the aesthetic sense. This aesthetic sense does not come automatically to all people with perfect vision and hearing, so it is fair to describe it as something extra, something not wholly reducible to vision and hearing. As the aesthetic sense informs us about what is beautiful, we can analogically understand the moral sense as informing us of what is good. People with a functioning moral sense get a clear impression of wrongness when they see (or perhaps even imagine) someone being mugged, for example.
However, though the wrongness is obvious, we may find it very difficult to list the features of the scene which account for the wrongness. We discover wrongness through observing natural properties with our five senses. Can we list the necessary and sufficient conditions such that any action which satisfies these conditions is wrong? The Ethical Naturalist thinks that in principle, we can. For naturalists, rightness and wrongness are nothing more than certain combinations of natural, non-evaluative properties. Since we can in principle build mechanical detectors for all these natural properties, the Ethical Naturalist thinks wrongness is something that a machine could eventually detect. The ethical intuitionist typically disagrees (although, it is not essential to the view): they see a wide conceptual gap between natural facts and evaluations. There seem to be no valid arguments in which purely descriptive/factual premises entail a prescriptive/evaluative conclusion. Ethical intuitionists claim that only an agent with a moral sense can observe natural properties and through them discover the moral properties of the situation. Without the moral sense, you might see and hear all the colors and yelps, but the moral properties would remain hidden, and there would be in principle no way to ever discover them (except, of course, via testimony from someone else with a moral sense).
For a recent criticism of sentimentalism (as a primarily metaphysical thesis), see François Schroeter (2006). D'Arms and Jacobson (2000) also provide a recent critique; however, they criticize "simple sentimentalism" and defend a more "sophisticated sentimentalism".