Emily is the ex-girlfriend of Maggie Sawyer, who broke up with Maggie after she caught her cheating on her.
Emily had a five-year relationship with Maggie Sawyer, but Maggie betrayed her, and when Emily knew about it she broke up with her, telling her that she didn't deserve to be happy. These words made a great impression on Maggie.
Years later Emily accidentally stumbled into Maggie and her new girlfriend Alex Danvers, and Alex convinced Maggie to ask Emily to go to dinner with them. Emily initially agreed, but eventually decided not to show up. When Alex went to ask her for an explanation, she told her that Maggie had betrayed her.
Emily seems to be a very correct individual; she vented a justified anger against Maggie Sawyer, telling her horrible things, but when she saw her again years later she tried to be civil with her, after revealing to Maggie's new girlfriend, Alex Danvers the betrayal of the cop, Emily seemed to feel guilty. This implies that she didn't want to sabotage Maggie's new relationship, and that even though she no longer wanted to deal with her, she didn't even want revenge.