As of 2015, Form 1040 is for individuals who make at least $100,000 in a single tax year, owe taxes for household employment and itemize deductions, while Form 1040A tax is for taxpayers who don't itemize deductions but can claim tax credits. There is also a Form 1040 EZ.
Form 1040 is also intended for individuals who earn money through unreported tips, self-employment, specific types of non-specific distributions and as a shareholder in an S corporation. The form is one of the more flexible and doesn't have the same boundaries as the Form 1040 EZ. The 1040 form is one of the longer forms and can be used by any taxpayer no matter what his filing status is.
Form 1040A is for taxpayers whose taxes are only derived from the tax table, an education credit recapture or the alternative minimum tax. The form allows only for a few credits and deductions. Individuals who can claim several deductions and credits may want to consider filing a Form 1040.
The Form 1040EZ is for those who are filing as married jointly or single and are under the age 65. The form is also for taxpayers who make less than $100,000 in a year and don't have any dependants to claim. It's best to consult the IRS website or a reputable and experienced tax accountant to determine which form to use.