We know little about the project itself. What we do know is that children with special powers were found and classified according to their powers by ranking them with leaves of clover. A one-leaf clover has a small amount of power, like Oruha who can see her death. A two-leaf clover has slightly more power, like Gingetsu. Those two groups are the most common levels. There were only three three-leaf clovers but one was killed by his brother, A. They are supposed to be the most powerful but there was one who had even greater powers. She was the four-leaf clover who was never supposed to exist.

The rest is speculation on my part. The project (which is my term for it) was most likely sponsored by the government. It's goal would be not only to categorize these people, but also to create a record of their identities and powers. Perhaps they had even more sinister intentions. Maybe they were hoping to be able to control those with the greater powers. Considering Lan's powers, that wouldn't be too inconceivable. However interesting this idea may seem, it's quite likely that the project was simply created to find those with the maximum power levels and eliminate them as a threat.

The question which comes to mind is whether people were aware of the project? Do they know what a "Clover" is? All of the Clovers have tattoos so do people understand why? All four of the Clovers we see have their clover hidden (Oruha has made her clover bigger which makes it look like a tattoo she chose). It might be that the Clover project is a secret only members and subjects know about or it might be something society demanded so that they wouldn't feel threatened by their powers, and now the Clovers must cover their marks to avoid detection. This, however, seems a little too X-Men for a Clamp story.