French wire is an extremely fine coiled wire. Imagine a toy Slinky but very, very small.
The purpose of this type of wire is to protect the thread between the last pearl in a necklace and the clasp.
In a French wire finish, after the last pearl has been strung, the thread is threaded through a short section of the coiled French wire and then looped through the jump ring of the clasp. Better quality pearl and bead necklaces are finished in this way.