If you are unmarried and lived together with the child, either one of you can claim the child (or children) as dependents. Typically, the best way to decide which parent should claim the child is to determine which parent has the higher income. The parent that earns the most income would receive a bigger tax break.
You can both create a 'hypothetical' return on TurboTax where each of you claim the child to see who gets the best refund possible.
UPDATE:
The Parent who decides not to claim the child should remove the dependent from their return. If not, this will cause a delay in both returns if the IRS has to determine who can claim the dependent.
