Income limits can vary considerably depending on how an individual qualifies as your dependent. Rules for your dependents are some of the more complex involved in tax law, but all dependents fall into one of two categories: they're either qualifying children or they're qualifying relatives. Income rules are much more strict for qualifying relatives.
Your qualifying child can be your biological or adopted child, a stepchild, foster child, or even a sibling or stepsibling. She can be the child of any one of these individuals, too. She must be younger than 19, or 24 if she's a full-time student, and she must live with you for more than half the year. There's no limit to how much she can earn and still qualify as your dependent, as long as she meets these other criteria. What she does with her income matters a great deal, however.
Your qualifying child cannot use her income to pay for more than half her own support. For example, if your teenager works a part-time job when she's not in school, she might earn $8,000 for the year. She might save half this toward college and use the balance for clothing and other incidentals. That's OK, because she's not supporting herself with the money. If she gives the $8,000 to you, however, and if you use it toward the mortgage, utilities and other necessary expenses of your household, the share of the household's expenses attributable to her must be at least $16,001 for the year, and you must pay $8,001 of that.
Qualifying relatives are typically people who don't meet the criteria as a qualifying child, but who either lived with you all year or who are related to you. A qualifying relative's income is limited to the amount of the standard exemption, $3,800 as of 2012. Social Security and other tax-exempt income does not count toward this limit. You must also pay more than half of your qualifying relative's support. For example, if she earned $3,000 in wages and had $8,000 in Social Security benefits, she would qualify unless she put all that money toward her own support and you did not also contribute $11,001 or more. Even though tax-exempt income doesn't count toward her income cap, it does count toward her support needs if she uses it for that purpose.
Claiming an earner as a dependent can also affect that person's own tax filing. She can only claim her earned income plus $300 as a standard deduction on her return if this amounts to less than the standard deduction for that year, $5,950 as of 2012. Your dependent cannot claim anyone else as a dependent on her own return. If your dependent is married, she can't file a joint return with her spouse unless they did so only to claim a refund.