MICHEL: You know what ennui is, yes?
LORELAI: Yes. Um, Webster’s defines ennui as a lazy, soon to be out of work, French concierge who won’t answer the phone.
Lorelai is referring to Webster’s Dictionary, which is any dictionary compiled by American lexicographer Noah Webster (1758-1834), or one named in his honour. His American Dictionary of the English Language was published in 1828, and his dictionaries helped redefine American cultural identity, and popularise American language and spelling. Webster’s was an important resource for poet Emily Dickinson, who read it religiously. The official dictionaries are published by Merriam-Webster.
Of course Merriam-Webster correctly defines ennui as, “a feeling of weariness and dissatisfaction”, and notes that it has been borrowed into English from French.