If you are the custodial parent where the child  physically lived for more than half the year (183 nights), then:

When you enter the dependent, you say that he/she is "Your child" (not you and your spouse if remarried), 
he/she lived with you the whole year, 
“no” the child did not pay more than half of his/her own support,
"yes", you have a custody agreement, 
and "yes", the other parent is claiming this year.  

That will give you the EIC, Child Care Credit and Head of Household filing status if you otherwise qualify.

The other (non-custodial) parent can claim the child’s exemption and child tax credit only and needs a signed 8332 form to do so.