Grand-mère vs. aïeule
While both grand-mère and aïeule can translate to grandmother in English, they are used in distinctly different contexts in French. Grand-mère is the common, everyday term, whereas aïeule is a more formal and historical term that often refers to a more distant female ancestor.
Grand-mère
A1This is the standard and most frequently used word for grandmother. It refers to the mother of one's father or mother and is used in all everyday situations, both formal and informal. The affectionate diminutive is mamie.
Ma grand-mère vient déjeuner avec nous dimanche.
(My grandmother is coming to have lunch with us on Sunday.)
J'ai passé toutes mes vacances d'été chez ma grand-mère.
(I spent all my summer holidays at my grandmother's house.)
Comment s'appelle ta grand-mère ?
(What is your grandmother's name?)
La recette de ce gâteau vient de mon arrière-grand-mère.
(The recipe for this cake comes from my great-grandmother.)
Elle est très fière d'être devenue grand-mère pour la troisième fois.
(She is very proud to have become a grandmother for the third time.)
Aïeule
C1This is a formal, literary, or historical term. While it can mean grandmother, it more often refers to a female ancestor or a foremother from a generation further back than a grandmother. It is rarely used in modern, spoken French to refer to a living grandmother and is typically found in genealogy, literature, or historical discussions. Its masculine counterpart is aïeul.
Mes recherches généalogiques m'ont permis de retrouver le nom de mon aïeule du 18ème siècle.
(My genealogical research allowed me to find the name of my 18th-century female ancestor.)
Le vieux médaillon appartenait à son aïeule.
(The old locket belonged to her foremother.)
Dans ce roman historique, l'héroïne s'inspire du courage de son aïeule.
(In this historical novel, the heroine is inspired by the courage of her ancestor.)
Il parlait de ses aïeules avec un grand respect.
(He spoke of his foremothers with great respect.)
Cette tapisserie a été brodée par une lointaine aïeule de la famille.
(This tapestry was embroidered by a distant female ancestor of the family.)
Summary
In essence, use grand-mère when talking about your direct, living grandmother in a normal conversation. Use aïeule when referring to a more distant female ancestor (a great-grandmother or beyond), or in a very formal, literary, or historical context. Grand-mère is modern and personal; aïeule is historical and ancestral.