Wiki offers, of The Enlightenment, The Age of Enlightenment (also known as the Age of Reason or simply the Enlightenment) was an intellectual and philosophical movement that dominated the world of ideas in Europe during the 17th and 18th centuries. The Enlightenment included a range of ideas centered on the pursuit of happiness, sovereignty of reason, and the evidence of the senses as the primary sources of knowledge and advanced ideals such as liberty, progress, toleration, fraternity, constitutional government, and separation of church and state. The page does offer a brief quote containing "project of Enlightenment" but I think that's not it.
More promising is The Enlightenment Project in the Analytic Conversation by Nicholas Capaldi one of whose chapters is The Enlightenment Project which says Alasdair Maclntyre, in his enormously important and influential book After Virtue (1981), identifies the ‘Enlightenment Project’ as the “project of an independent rational justification of morality”... we use the same expression as Maclntyre, namely ‘Enlightenment Project’, and while we agree that part of that project was to establish the authority of Judeo-Christian morality by reason alone.... So already there's a slight disconnect: was it to establish a morality, or was it to establish the morality that everyone already knew was correct?
How has this project got on? Pretty well, I'd argue. We don't all agree on all details of morality, but we do rather largely agree that our morality isn't grounded in religion. Actually I'm pushing that too far: I think many of the religious - who might even represent a majority in the USofA - would say that their morality is grounded in religion; but then we face the bizarre coincidence that those of us who are good atheists have essentially the same morality. You can try to get round that by saying you morality comes from your upbringing, and your parents were believers, but I don't really buy this. Let's poke around in the Ten Commandments, not caring about the numbering too much since people seem to differ on even that, and ignoring the "I am the Lord" type ones which I think everyone can agree can't be grounded in rationality, we get: Thou shalt not murder / Thou shalt not commit adultery / Thou shalt not steal / Thou shalt not bear false witness against thy neighbour / Thou shalt not covet thy neighbour's house or his wife or his slaves, or his animals, or anything of thy neighbour. If you then drop regrettable bits like implicitly condoning slavery, which rational secularism definitely doesn't; and making the wife in the same class as property of the husband like his animals, ditto; then we end up with a set that you can trivially justify rationally - I wave my hands here as to exactly how because that isn't my current point. And I'll add that a couple of them don't come through so strongly: against adultery, or honouring-they-father-and-mother. Adultery, I would rephrase as not-subverting-bonds, which makes it more justifiable (actually I think I barely need to do that; adultery is immoral, but may not be illegal, but that's different). HTFAM is harder; I notice it sits between the Lord ones and the Normal ones; wiki points out that they were enforced as law in many jurisdictions, and are still considered enforceable law by some; it also notes the connection with honouring god.
Given all that, why does JG believe that our age is "distinguished by the collapse of the Enlightenment project on a world-historical scale"? He continues with "shed their traditional allegiances and their local identities and unite in a [sic] universal civilisation grounded in generic humanity and and a rational morality..." And it seems that what he is sad about is "renascent particularisms, militant religions and resurgent ethnicities" (bear in mind this was written in 1992). And yet the secular West remains strong.
So his (implicit) defn of EP rather appears to contain a lot of practical politics, but I think he is over-pessimistic. I'm also somewhat doubtful that it can be meaningfully categorised as a "project", but that's a different matter.
Update: having talked to Mfd, I think the subset I picked - establishing morality by reason - was too narrow. I should probably have stuck with Wiki's version; and the distinction between EP and E is probably spurious; the modifier P is both unnecessary and confusing, in that it suggests a concerted planned effort that did not exist.
Another Update: from essay 6 (Agonistic Liberalism) we have "sought to found political authority on the rational choice of its subjects rather than on tradition or local prescription" which is nice.
* On Morality (2008).