In US parlance, a lost and found is a lost property office, where people who find lost items can hand them in, in the hopes that the owner may look for it there. Usually found at airports, train stations, amusement parks etc. The lost and found system dates back to medieval Japan, but the first modern lost and found office was organised in Paris in 1805, under Napoleon.
This episode revolves around some lost property which has been found, and will be handed in for the owner to retrieve it.
The Annotated Gilmore Girls 