The alternative minimum tax is a second way of calculating your federal tax. It starts from your income, adds back certain deductions and preferences, subtracts an AMT exemption, and applies rates of 26% and 28%. You compare the result to your regular tax and pay whichever is higher. For most salaried people the regular tax is higher and AMT never comes up.
The main way ordinary people meet AMT is through incentive stock options. When you exercise ISOs and hold the shares, the spread between strike and market value is not regular income, but it is AMT income. Exercise 20,000 options at a $3 strike when shares are worth $18, and you have added $300,000 to your AMT calculation. That can produce a tax bill of $70,000 or more in a year when you sold nothing.
Some of that AMT comes back. The amount paid because of ISO exercise creates a credit that can offset regular tax in future years, though recovering it can take a long time.
The common mistake is exercising a large block of ISOs in December without running the AMT numbers, then discovering the bill in April with no cash to pay it. Exercising early in the year gives you until the following April to see the full picture, and exercising in smaller lots each year can keep the spread under the AMT exemption entirely.