The feeling you wake up with in the morning after an emotional breakdown from the night before. This feeling is sometimes a result from an argument, a deep sadness, an unfortunate loss, a break-up, drama, strong feelings of regret, conflicting emotions, etc.
Typically the night before, you have trouble sleeping - you may even be physically tired yet remain mentally awake - which might involve tossing and turning all night long, a constant vigilance of the clock, and attempts to read or watch TV to try and sleep quicker.
Instead of sleeping away your sadness and waking up refreshed, the feelings come back to make you feel bad, maybe even worse. Now that you've had time to dwell on the situation and ponder your own feelings, repeating those feelings again the next day makes it sting a bit more.
Sometimes an emotional hangover is followed by ambivalence(conflicting emotions), stress, more drama, and/or just a really bad mood.
This is bad especially if you aren't a morning person. Usually this feeling comes to you when you least expect it: when someone mentions the situation to you, when you go online to check your mail, listen to some music, on your way to work/school, etc.
Guy: Hey, you don't look so good...rough night?
Girl: Yeah, I got into a fight with my ex last night and now I'm having an emotional hangover.
Guy: OH, I hope you feel better. Ya know, I'm still available ;)