The following is taken from the Texas Family Code.
§ 153.131 (b): IT IS A REBUTTABLE PRESUMPTION that the appointment of the parents of a child as joint managing conservators is in the best interest of the child. A finding of a history of family violence involving the parents of a child removes the presumption under this subsection.
What does this mean?
It means that the court MUST appoint both parents as joint managing conservators unless (1) they have a written agreement otherwise or (2) there is a finding of family violence.
Typically, one parent will be declared the Primary Managing Conservator, meaning that the child lives with that parent, and the other parent will be declared the Possessory Managing Conservator. The rights and duties of BOTH parents are very much the same. Consult the attached chart for a further explanation.