Child tax

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.

@Kionalew 
Great question!
If someone else claims your child when they were not supposed to, and if you know that they are filed their tax return, then you would need to mail in your completed tax return. If you e-file your return, it will automatically be rejected because someone else claimed a dependent on their tax return.
You would want to file your return on paper and claim the appropriate children on your return.
This typically means that you will need to "paper-file" your return.
When you file with TurboTax, you will be given specific instructions on what to mail, as well as where to mail your return to.
Once the IRS does process your return, which can take longer than normal, you may receive a letter from the IRS letting you know that a child was claimed on another return. The letter will usually ask you for more information about the child you are claiming. 

Normally, the other person that is claiming the child will also receive a letter, asking for proof that they are able to claim the child.  Normally, the letter will ask someone to amend their return.  If you do receive a letter from the IRS, make sure that you reply to that letter as soon as possible.

Even though their may be more work on  your end, the end result should be beneficial to you.
Thanks.