I didn't quite get from the first episode what happened to Ohana's family and why her parents had to "run away" somewhere, sending Ohana to live in her grandmother's hotel. Why did this happen? Where are her parents now?
Ohana's biological father - Matsumae Ayato - died when Ohana was still a baby, of unspecified causes (probably a Japanese cold or something). I believe this is only alluded to in the show, but if you watch the movie, you actually get to see that stage of Ohana's life.
After that, Ohana's mother (Satsuki) gets in relationships with various boyfriends, the latest of which is the guy you see in episode 1. The dude apparently is in a lot of debt - Satsuki has poor taste in men. Worse still, when his creditors asked him for his address, the dude seems to have given them the address of Ohana's apartment.
Rather than dealing with his debts like a reasonable person, he and Satsuki decide to abandon the apartment and run away together so as to get the creditors off their trail. To them, this doubles as a romantic getaway, and thus, they don't want to take Ohana along with them. Satsuki decides to ship Ohana off to Kissui, and thus the show begins.
Basically, it all boils down to Satsuki being a godawful excuse for a parent.