The non-custodial parent signs Form 8332 and sends it in.
The Form 8332 can't be e-filed. The non-custodial parent must have signed the form and then mail it to the IRS along with the transmittal Form 8453 within 3 business days from the date the return is accepted. The 1040 income tax return can still be e-filed. On the 8453 you simply check the box Form 8332, Release/Revocation of Release of Claim to Exemption for Child by Custodial Parent.
The forms should be mailed to:
Internal Revenue Service
Attn: Shipping and Receiving, 0254
Receipt and Control Branch
Austin, TX 73344-0254
Are you the custodial parent? Do you have an agreement with the other parent to allow the other parent to claim them--due to divorce or that you live apart and share custody? Did one of you sign a Form 8332?
If there is a signed 8332 then the custodial parent retains the right to file as Head of Household, get earned income credit and the childcare credit. The non-custodial parent gets the child tax credit for children under the age of 17.
As far as the IRS is concerned, the custodial parent is the one with whom the child spent the most nights during the tax year--at least 183 nights.
If you are a non-married couple who live together then only one of you can claim the child(ren) and the one not claiming the child does not enter anything at all on their tax return about the child.