How to say
Happy Mothers Day in Turkish
Happy Mother's Day is
Anneler Günün Kutlu Olsun.
Anne is mother or mom.
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
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.