I need to make assumptions for a typical situation because all the details are not provided. I am assuming that you are recently divorced, legally separated, or you lived apart all of the last half of the year. From your statement, it appears that the child lived exclusively with you and not the father. In such cases, you would be the "custodial" parent, and the father would be the "non-custodial" parent.
Note that the determination of "custody" is based on with whom the child lived. You don't trade that back and forth unless the child physically lives with each parent more than half time on alternating years. Basically, if you and the father are supporting the child, and if the child ALWAYS lives with YOU for over half the year (and the child never switches arrangements to live with Dad for half of any year) then you would ALWAYS be the "custodial" parent because you retained the custody of the child.
So if you are always the "custodial" parent for every year, then every year (whether or not the father gets the dependency for the child) you would always (assuming you are not remarried) get the option to file Head of Household, and you would always get the earned income credit for the child, and you would always the get any child and dependent day care credits (assuming you are otherwise qualified) to provide for care that allows you to work. The father would NEVER get those benefits, whether he had the dependency or not, because the father would be the "non-custodial" parent.
In the years where the father does claim the dependency, then the father does get the child tax credit and education credits, if any. In the years where the father does claim the dependency, then when you file your own tax return, you must list your child on your return and you would need to click a box to indicate that the listed child is being claimed as a dependent on someone else's return in order to avoid two taxpayers claiming the same dependency.
If you take the dependency, then then the father should not take the dependency. If both you and the father claim the dependency, then the second return to get filed will be rejected. When a tax return is rejected, the IRS will then look at both returns and the correct legal situation should be determined and corrected. (Try to get it right the first time, and save yourself a lot of work, and avoid the risk of possible fraud determinations.) You should follow the legal requirements of your separation agreement or divorce documents. In order to give the father the dependency for the child, you would normally give the father a Form 8332 to grant the dependency to him. You can issue a new 8332 to remove his option to take the dependency, but I recommend that you talk to your divorce lawyer before taking such a step because you do not want to be the one to violate any legal agreements that may be in place.
