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AMT stands for Alternative Minimum Tax. The Alternative Minimum Tax was developed in 1969 to make sure that wealthy taxpayers didn't escape paying income taxes. The tax was meant to target high-income taxpayers who may have many deductions and can sometimes avoid paying any income taxes at all.
To make sure that all taxpayers with substantial income are not able to avoid paying tax, the law limits the benefit a taxpayer can receive from favorable treatment of certain deductions and preferences.
The Alternative Minimum tax is computed on IRS Form 6251.
If the Alternative Minimum tax calculation results in a higher tax then the Regular income tax, then the difference is added to the Regular tax computation.

If the Child Tax Credit is larger than your tax liability, the unused part of the Child Tax Credit can be refundable as an additional Child Tax Credit.
The portion that is available for a refund depends on the number of children you have, your total earned income, and, sometimes, the amount of taxes you have paid over the year to Social Security and Medicare.
Use Form 8812 to claim the Additional child Tax Credit.

Taxpayers who make under the $110,000 / year are currently still allowed to claim $1,000 per child as their Child Tax Credit. Families making between $110,000 and $130,000 receive a reduced credit (specifically $50 per $1,000 made over $110,000).

They must not have reached the age of 17 by the end of the tax year.
Children may be your blood children, grandchildren, step children, and / or adopted children. Foster children also fall under this category as long as they have lived in your home for the entire year in question.
Qualifying children must be citizens or residents of the United States who can be claimed as dependents by the taxpayer.

Only those families that make less than $110,000 annually are eligible for the full credit. Families making between $110,00 and $130,000 receive a reduced credit (specifically $50 per $1,000 made over $110,000). Individuals who are unmarried must have earned under $75,000 to qualify for this credit. Taxpayers who are married but filing separate returns must have made no more than $55,000.

A qualifying child for purposes of the child tax credit is a child who is your son, daughter, stepchild, foster child, brother, sister, stepbrother, stepsister, or a descendant of any of them (for example, your grandchild, niece, or nephew) and
was under age 17 at the end of the current year,
Did not provide over half of his or her own support for the current year,
Lived with you for more than half of the current year
Is claimed as a dependent on your return,
Does not file a joint return for the year
Was a U.S. citizen, a U.S. national, or a U.S. resident alien.

Contributions designated under section 414(h)(2) are treated as employer contributions and as such they are not voluntary contributions made by the employee. They do not qualify for the credit and should not be included on Form 8880, line 2.

You cannot take the credit if you are a student. You were a student if during any part of 5 calendar months of the current year you Were enrolled as a full-time student at a school, or Took a full-time, on-farm training course given by a school or a state, county, or local government agency. A school includes technical, trade, and mechanical schools. It does not include on-the-job training courses, correspondence schools, or schools offering courses only through the Internet.

You cannot claim the credit if the person(s) who made the qualified contribution or elective deferral (a) was born after January 1, 1994, (b) is claimed as a dependent on someone else’s 2011 tax return, or (c) was a student.

You are not eligible for the credit if your adjusted gross income exceeds a certain amount. You cannot take the credit if either of the following applies: The amount of AGI on Form 1040, line 38 ; Form 1040A, line 22; or Form 1040NR, line 37, is more than $28,250 ($42,375 if head of household; $56,500 if married filing jointly).