Besuel was Rivka's father and Avraham's cousin. When Eliezer was sent to look for a wife, he met Rivka who not only gave him a drink, but quenched the thirst of his ten camels (no eay task even for ONE camel.) Since she seemed like a nice girl, and was related to Avraham's family, Eliezer went to her house to finalize the marraige.
And here's where the Midrash gets in the way.
According to the p'shat, Besuel's family gives Rivka up without a fight, simply saying, "OK, if that's what G-d wants, we have no argument. But let's get Rivka's permission, since, it is, after all, her life." (Pretty enlightened for Mesopotamia circa 3,000 years BCE). However, they do want a few days to say goodbye, which, again, does not seem unreasonable since she's going to be moving from Mesopotamia to Israel and the invention of Skype is still several millenia off. However, according to Rav Becahye and the Baal HaTurim, Besuel is not the kindly father who permits his daughter's marriage to his cousin's only legitimate son. He's EEEEVIIIILLLL! We're talking murderer-evil! We're talking trying to poison Eliezer so Besuel could rob him evil. However, the
Wow. Here we have a family that has a daughter nice enough to water ten thirsty camels--without being asked--and she's the spawn of murderers! And they seemed like such nice people!