If you are married filing separately and both parents live with the child, either parent can claim the child as a dependent and only they get all the child related benefits ... they cannot be split.
If you are married and did NOT live together the last 6 months of the year, live apart and share custody, then:
The parent with whom the child lives more than half the year (183 or more nights ) is automatically entitled to claim the child as a dependent. This is the custodial parent. (IRS determines custody based on where the child lives, not any court order or agreement.)
The non-custodial parent is not entitled to claim anything.
However, the custodial parent can sign a release (form 8332) allowing the non-custodial parent to claim the child as a dependent. You can download this form from the IRS web site. The custodial parent signs it and gives it to the non-custodial parent and the non-custodial parent mails it to the IRS after e-filing the rest of their tax return. In this case, the non-custodial parent can claim the dependent exemption and the child tax credit. The non-custodial parent can never claim earned income credit, the dependent care credit (day care credit) or use the child to qualify for head of household status. Those benefits always stay with the custodial parent.
