Generally, day to day decisions about the bringing up of a child are made by the parent with whom the child lives. Bigger decisions are made by the parent who has what the law calls 'parental responsibility'.
A parental responsibility order can be made by the court if there is no agreement. There can also be 'specific issues orders', which relate, as the name suggests, to specific things, like for example what school the child goes to, what religion he/she is brought up in, and so on. And there can also be things called 'prohibited steps orders', like for example, prohibiting a parent from taking the child out of the jurisdiction, or whatever it may be.