Each person is allowed a deduction for personal exemption. This amount is $3,800 in 2012 adjusted for inflation. What this means is that if a person has $3800 of taxable income (less deductions), she or he pays no tax.
In addition, a taxpayer is allowed an additional personal exemption for each dependent. A dependent is a person with certain specified relationships to the taxpayer and with more than half of whose support is provided by the taxpayer. Children cease to be dependents, even if supported by a parent or parents, in the year in which they attain age 19, or in the year in which they attain age 24 if they are full-time students for at least 5 months during the taxable year. The definitions of "support" and "full-time" result in many disputes and persons claiming these exemptions should consult with a tax adviser.
There are rules dealing with a dependent who is supported by several taxpayers. Essentially, any person paying at least 10% of the support can claim the exemption if all other 10% or more payers agree (and none of them paid more than 50% of the support) and have the appropriate relationship to the claimed dependent.
The personal exemptions were, like many other tax benefits, phased out for taxpayers with substantial taxable incomes. The phase-out limits are not applicable in 2010, 2011 or 2012, but are resurrected in 2013.