The subject of marriage first comes up in Week 5 when Talarist, the Duke of Sedna sends Elodie a letter of condolence for her mother's death, along with a necklace. Wearing the necklace represents a tacit agreement to a marriage alliance. You can avoid this alliance by having Court Manners >10 and deciding that wearing a man's gift is inappropriate (+1 Yielding). If you wear the necklace, it's still possible to break off the engagement when Talarist shows up for a visit.
The second time marriage is brought up is in Week 9, when Banion comes to request Elodie's intervention in the conflict with Ixion. If Elodie is wearing her betrothal necklace he will question her about it, otherwise he will flatter her and she will wonder whether he's trying to flirt with her. During the negotiations with Ixion, it's possible to resolve the conflict by ordering Brin, Banion's sister, to marry an Ixionite and provide heirs who will put an end to the grudge. Doing so unlocks two events later on, and will cause Joslyn, Elodie's father, to start seeking out a second wife.
During the Royal Ball (Week 17), Elodie has the option to dance with a number of different suitors. Dancing with a partner who's already married, or someone "scandalous", is supposed to send the message that Elodie is not prepared to pick a husband yet. If Brin is newly married, Banion will attempt to insinuate that he is having an affair with Elodie, and agreeing to dance with him confirms this, situating him as her betrothed.
If Elodie danced with him the Royal Ball (Week 17), Linley of Kigal becomes a potential suitor at the end of the game.
In Week 22 Elodie receives news of the death of Adair's father. He is too young to rule, and one of the ways you can resolve the succession dilemma is to bring him to the royal palace as the princess's betrothed. Adair is very young and the marriage would not take place until years after Elodie's coronation, but having a high score with animals helps smooth over the engagement, since he takes an interest in frog-catching. Otherwise, he remains a little scared of Elodie.
If Elodie has gone on an adventure with Briony (Week 29) and saved her life, before the coronation Briony will send her an admiring letter. Briony will appear on the list of prospective suitors and choosing her will cause her to be appointed lady-in-waiting. Although both Elodie and Briony end up marrying unnamed noblemen, the epilogue states that they are in love.
If Elodie has romanced a commoner in weeks 6, 14, & 28 she can marry Evrard the commoner.
Kevan is listed as a marriage option at the end if he survives to the end of the game, but trying to marry him will not be successful unless the family secret was discovered in that playthrough.
Ignatius is listed as a marriage option if he divorces his wife and Elodie finds out about it. Ignatius will reject her (and Elodie will move on to a dashing naval captain) unless she resolved things peacefully with Togami.
If Elodie remains unmarried she may get an heir with an unnamed father, whose identity remains a mystery. She may also choose to marry a foreign duke or an earl who likes pastry.