Esperanza used to be very rich and Miguel was like her servant. In California they are both very poor and Esperanza is kind of debted to Miguel for his help. The deep river between them is basically a "barrier" because Esperanza felt she was so different from Miguel. She thought she was a princess and he was her servant.
The entire quote is: But now that she was a young woman, she understood that Miguel was the housekeeper's son and she was the ranch owner's daughter and between them ran a deep river. Esperanza stood on one side and Miguel stood on the other and the river could never be crossed. (2.52). This means that she realizes that Miguel the son of the housekeeper and that because of who her family are that they will never be allowed to be together.
what esperanza means is since miguel is a servant she can never go out with someone that is a servant because they are poor and she is rich. in california this changes because esperanza becomes poor and then she sees how bein poor is not so bad it is mostly the same as baing rich.