With all the deductions, credits, and other shenanigans in the tax code, it has become possible for more and more people to end up paying nothing in taxes even though their income is relatively high.
This is not good from a revenue point of view. The AMT is a way for the government to decide the discounts you got were too many and that you indeed to owe tax money.
The issue is of relevance today because of higher home ownership and other offsets, more and more people are subject to the alternative minimum tax than ever before. If you manually file your paperwork, be aware.
If you make more than $75,000 a year and have some large deductions, you might be subject to the tax.
Most (if not all) reputable accountants or filing software will calculate the need for you. Be sure you know, because not paying when you should means paying penalties if you are audited. The IRS does not forget, nor does it forgive.