The IRS looks at who the custodial parent is and who is not. The custodial parent is who the child lives with for more time during the tax year. The states follow the same rules.
When you allow a non-custodial parent to claim a child, you are only releasing the ability to claim the child for Child Tax Credit under the special rule for divorced and separated parents.
The custodial parent retains tax benefits related to Head of Household filing status, Earned Income Credit, and the Child and Dependent Care Credit. To be eligible for these credits, the child must have lived with you for more than half the year, so the non-custodial parent is not eligible for those credits.
This article will give you more information: Divorced and Separated Parents