If a court grants you (or your former spouse) custody of the children, the other parent (or you) will be given the right to visit, unless the court believes that the noncustodial parent's physical presence would be detrimental to the children. As with custody, decisions about visitation are also made based on the "best interests" of the children. If you and your ex-spouse can communicate enough to agree on a visitation schedule, the court will probably grant "reasonable visitation rights" and leave it to you to work out the details as to when, where, and how visitation will take place.
If, however, you and your ex-spouse cannot agree on visitation, the court will define the visitation rights for you. The court might say, for example, that "Barbara Washington shall have the right to visit with the children every Saturday from 10:00 a.m. to 5:00 p.m., plus three weeks during the summer months, the weeks to be agreed upon by the parties"; or "Sean Washington shall have the right of visitation on the first weekend of every month from 6:00 p.m. on Friday to 6:00 p.m. on Sunday provided he picks the children up from, and returns them to, Barbara Washington's home."
Sometimes, courts impose restrictions on visitation beyond times and places, such as requiring 24-hour notice of visitation; prohibiting the noncustodial parent from removing the child from the county, state, or, in rare cases, the child's own home; prohibiting the noncustodial parent from drinking alcohol while with the children; or where there's evidence of abuse, requiring another adult to be present during the visit.
In addition, courts in a few states may prohibit a parent from visiting the children in the presence of the person that parent lives with, or prohibit the children from spending the night with a parent whose new partner is present in the house. If the custodial parent has a new relationship, most courts will continue to allow visitation as long as the living together situation doesn't have an adverse effect on the child. Louisiana is one state that still prohibits visitation when the parent's live-in lover is present in their home, even when the couple is discreet, because such visits might undermine "the children's respect for the family institution." (Lasseigne v. Lasseigne, 434 So.2d 1240 (La. 1983).) Court decisions like this are far from the rule, however, and if you've been given such an order, speak to an attorney regarding your obligations, your rights, and a possible appeal. Check Nolo's Lawyer Directory for names of local family law attorneys. But in the meantime, understand that it's not wise to violate a court order.