Madame Forestier is the old friend of Madame Loisel in this sharply ironic short story by Maupassant. We are not told much about her but we do know that she is wealthy and therefore has a stock of jewels that her friend can borrow. She seems almost to be careless of her wealth. Consider how she lets Madame Loisel choose what she wants:
Mme. Forestier went toward a large closet with mirrored doors, took out a large jewel box, brought it over, opened it, and said to Mme. Loisel, "Pick something out, my dear."
She is certainly very open with her possessions, and gives Madame Loisel free reign of her jewels as she looks for something for the party.
When they return the (new) necklace, Madame Forestier is quite sharp with Madame Loisel:
When Mme. Loisel took the necklace back, Mme. Forestier said to her frostily, "You should have brought it back sooner; I might have needed it."
Although she speaks sharply here, it is with sympathy that Mme. Forestier addresses Mme. Loisel at the end of the story and tells her of her mistake:
Mme. Forestier, quite overcome, clasped her by the hands. "Oh, my poor Mathilde. But mine was fake. Why, at most it was worth only five hundred francs!"
Mme. Forestier therefore is a character who is a friend of Mme. Loisel's who lends her the jewels, appears slightly put out when she returns them, and then finally embraces her with sympathy because of the mistake she has made.