How to say Happy Mothers Day in Turkish

Happy Mother's Day is Anneler Günün Kutlu Olsun.

Anne is mother or mom. Anneler is mothers (that's plural, not possessive—odd, I know). Anneler Günü is Mothers' Day. Finally, with the n suffix, it becomes Anneler Günün which becomes your Mother's Day. Kutlu olsun simply means may [something] be happy, so the full literal translation is may your Mother's day be happy.

If you're conveying these wishes to your partner's mother or grandmother with whom you might have to be a little bit more formal, you might be better off using the formal you. I.e., Anneler Gününüz Kutlu Olsun.

Ana also means mother, but it's less commonly used in urban areas. No one will ever say Analar Günün Kutlu Olsun, but you might hear a lot of cennet, anaların ayakları altındadır which is a hadith. Your partner's family may be very traditional without being very religious. You're always better off starting out by being more deferential than you're used to until you figure out the maternal relationships in that particular family.

In group conversations/discussions (including online), it is customary for the male members of the group to say something along the lines of bütün annelerin anneler günü kutlu olsun, which just means happy mother's day to all mothers.