If the father is claiming the child this year but the child stayed with you the entire year, Do you remove the child from your taxes? & if you dont what's the point in him filing?

Wait---as far as the IRS is concerned,  the custodial parent is the one who can claim the child.  Do you have some sort of court order that says you have to allow the other parent to claim your child?  The custodial parent is the one with whom the child spent the most nights--at least 183 nights.

The custodial parent can sign Form 8332 that will allow the non-custodial parent to get the child tax credit.  The custodial parent can still file Head of Household, get earned income credit and the childcare credit.

As the custodial parent you must list the child in the program then follow the screens to indicate you are waiving that exemption to the non custodial parent via a form 8332 ... so if you are eligible for the EIC or day care credit you will get them.

IRS rules on this matter:  

The custodial parent has first priority on claiming the children on their taxes; regardless of the amount of support provided by the non-custodial parent. The non-custodial parent can only claim the child as a dependent if the custodial parent gives permission (on form 8332) or if it's spelled out in a pre 2009 divorce decree. . Even if a divorce decree, dated after 2008, gives the non-custodial parent the right to claim the child, they must still get form 8332 from the custodial parent. A properly worded decree should require them to provide that form.

For tax purposes, there is no such thing as shared custody, regardless of what your legal agreement says. The requirement, to be custodial parent, is that the child live with you MORE than 50% of the time. One of you has to be the custodial parent and the other the non-custodial parent.

Even if the custodial parent has no income and cannot claim the child, someone else in their household would have a higher priority than the non custodial parent on claiming the child. Usually it's the grandparent or other family member.

 There is a special rule in the case of divorced & separated (including never married) parents.

When the non-custodial parent is claiming the child as a dependent/exemption/child tax credit; the custodial parent is still allowed to claim the same child for Earned Income Credit, Head of Household filing status, and day care credit. This "splitting of the child" is not available to parents who lived together at any time during the last 6 months of the year; then only one of you can claim the child for any tax reasons. The tax benefits may not be split in any other manner.

Note in particular that the non-custodial parent can never claim the Earned Income Credit, Head of Household filing status or the day care credit, based on that child, even when the custodial parent has released the exemption to him.