The Millennium Falcon originally belonged to Lando, Han won the ship in a card game ([Sabacc][1]). It's never made clear if Han cheated or won it legitimately, but it's insinuated that Lando wasn't too happy about it, and might be holding a grudge from it.

>During the final round of the Cloud City Sabacc Tournament, Calrissian threw in a marker for "any ship on his lot" (at the time, he was the owner of a used spaceship lot on Nar Shaddaa, where the Falcon was parked). Solo, the other contestant remaining in the tournament, accepted the marker, knowing that Calrissian had taken a cruise ship to the tournament and left the Falcon behind. When Solo won the game, he claimed the Falcon as his own on the technicality which had been overlooked by his unlucky opponent.

>-http://starwars.wikia.com/wiki/Millennium_Falcon


  [1]: http://starwars.wikia.com/wiki/Sabacc