If you and your partner live together, and you are both the parents of the children who live with you, then you both qualify to claim them and can decide between you, for example, to each claim one child on your separate tax returns. List only the child you are claiming on your tax return. If a child's SSN appears on more than one return, the second return will be rejected.
The tie-breaker rules mentioned by @xmasbaby0 apply in the situation where the parties can't come to an agreement.
See this IRS webpage for more information on who qualifies for the Child Tax Credit.
To file as head of household, you must furnish over one-half of the cost of maintaining the household for you and a qualifying person. Therefore, only one of the parents will have contributed more than one-half of the cost of maintaining the household and be eligible to file as head of household. See here for more information. The other parent will file as Single.
@ryan791
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