A child can't provide her own care, so she definitely counts as someone's dependent come tax-time. If you and your boyfriend meet a few requirements, he can claim the child as his dependent. However, if you qualify to file your own taxes, you must claim the child since it's your biological offspring.
Generally speaking, if your child can be claimed by another parent, like his father, your boyfriend may not be able to claim the child on his tax return. However, if the father is not involved in the child's life -- and pays nothing to support the child -- and the three of you live together, and your boyfriend supports you both, there's a chance he can claim your child on his tax return.
In order for your boyfriend to claim your child on his taxes, the three of you need to meet a few requirements. First, you must have made $3,650 or less for the year, thus qualifying as a dependent yourself. You and your child need to have lived with him for the full year, for example, all of 2010. Your boyfriend must have provided over 50 percent of the child's expenses, from food to medical care to shelter. Since the child can't care for herself, these conditions are fairly easy to meet.
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Head of Household
Although your boyfriend can claim the child if the support conditions discussed above are met, he cannot claim "head of household" status, since neither you nor your child are related to him. If the child is biologically his, however, he can claim "head of household" status.
Before you have your boyfriend claim your child, make sure the child's father is not planning on claiming the child. A child can only be claimed once by one parental figure. If both of you claim the same child, whoever filed first typically gets to keep the child as a dependent, so if your boyfriend files second, he'll have to redo his taxes without the child dependent.