In good years we don't know what to do with them all. This year, and thanks to a series of house-guests, they have all been eaten fresh from the tree. Last year I made fig jam (you need very little sugar) and bottled the puree, without sugar, for eating with cheese and meats:very good! Single peeled figs wrapped in a thin leaf of prosciutto crudo is a very delicious appetizer; quartered but left to stand on a bed of leaves with pieces of fresh pecorino (sheep's milk) cheese and a little thyme, some honey drizzled on top, is another lovely way to go, taught me by an Australian friend who visited recently.
In the past, I am told, the bottom of the burgeoning figs would be painted with a little donkey urine to keep away insects and blight; although our figs receive no such treatment we still cut off the base before eating. I'm not sure if it's because there might be insects, or because we're worried someone might have treated them on the sly!
The Italians have an elegant way of eating figs: since they generally avoid eating the skin, they hold the fig by its stem, cut through the base to divide the fruit almost in quarters; the result is an opening flower. They then bite off each quarter, peeling it off the skin as they go. The skin is discarded.