The concept of custody can be separated into two (2) categories: legal custody and residential custody. Legal custody is the the authority to make major decisions about the minor child. Major decisions are those having a significant impact on the child and relating to the child's health, education or religion. Changing schools, having elective surgery or changing churches are examples of such major decisions.
Legal custody can be granted as sole or joint custody. If a parent has sole custody, that parent makes all major decisions on her or his own. If the parents have joint custody, they have to agree on the major decisions about their minor child. If they cannot agree, Illinois law requires them to mediate. If mediation fails, the court will decide based on what is in the child's best interest.
A variant on the typical joint versus sole custody arrangement is where one parent has sole custody, but agrees to not alter something without the other parents permission. For instance, the sole custodian might agree to raise the child in a particular religion, and not change the church without the other parents written consent. Alternatively, the parents may have joint custody, but one of the parents is given much more decision making authority.
Residential custody is sometimes called physical custody. This refers to where the child resides primarily. The parent with whom the child resides primarily is often called the residential parent or the custodial parent. The other parent has parenting time or visitation, and is often called the non-custodial parent. The visitation schedule can be anything the parties agree to (and the court approves), and failing agreement, whatever the court orders based on the best interest of the child.
Parents can share custody. In a shared custodial arrangement, the parents are often live close to each other, even in the same school district, and the child lives about half the time with each parent. The child might live with each parent for two (2) weeks at a time. If the parents live far apart, the child might spend longer in each residence, perhaps as long as six (6) months at a time with each parent. Shared custody is not typical, but has become more common in recent years.
There are variants of the above possible. Everything depends on the particular individuals involved. What is very important to remember in all cases is that having sole custody has nothing to do with the other parent's access to the child or day-to-day decisions affecting the child. You can have authority over all the major decisions retarding your child, but the other parent may have extensive parenting time, bordering on shared residential custody even. Likewise, regardless of legal custody, each parent makes day-to-day parenting decisions while the child is with that parent.
You can read the relevant statute here for yourself.
PLEASE NOTE: the above is NOT legal advice. It is only a general statement about the state of the law. It is not a recommendation and should not be used to make decisions regarding you specific situation. For such a recommendation, consult an attorney regarding your specific factual circumstance.