This is the former romantic relationship between the former Enhanced Original vampire/current human Alaric Saltzman and the vampire Caroline Forbes. Their relationship started off as a non-romantic and non-friendship teacher-and-student relationship and has developed on a friendship basis throughout the series. After Alaric's former fiancée died while being pregnant with Alaric's and her twins the Gemini Coven found a way to transfer the twins into Caroline and she gave birth to them. Caroline started loving the twins as they were their own children and helped Alaric raise them. Thus Caroline and Alaric became much closer and perhaps developed deep romantic interest which is indicated by the two getting engaged some time between the flashforward that took place in Season Seven.
Eventually Alaric fell in love with Caroline and proposed to her. Although Caroline does not return those feelings, she still said ‘yes’ because it would be the best for Lizzie and Josie.
However, Alaric broke up with Caroline as of the end of Season Seven, despite still being in love with her. This ends their engagement to each other.
In the premiere of the eighth season, Alaric suggests that they live together being they’re both struggling with their personal obligations and raising their twin daughters at the same time. Whether or not Alaric suggested this because it would make life easier for them and their daughters or if he was still in love with Caroline, she knew it wasn’t a good idea and she told him this, rejecting his inquiry of moving back in together. Due to the sirens’ involvement with Mystic Falls, both Caroline and Alaric struggled to keep their children safe during Season Eight. After the sirens and Cade were finally defeated; Caroline and Alaric, along with Alex, opened a school for supernatural students in Stefan and Damon Salvatore’s old home and they are currently working together while raising their siphoner-daughters that go to that school as well.
As of Legacies, Caroline and Alaric are raising their children separately. With Caroline in Europe recruiting for the boarding school, Alaric must run their school by himself. However, Alaric is also tasked with raising Lizzie and Josie while Caroline deals with other situations outside of the boarding school.