The most common scenario that impacts the AMT is an exercise and hold of ISOs. If you hold the shares past calendar year end, the bargain element (or the difference between the FMV at exercise and the strike price of the ISO, multiplied by the number of options exercised) is an adjustment on form 6251 when figuring the AMT. If the bargain element is enough to have the TMT exceed the regular tax, then AMT may be due.
The sale of shares acquired through an ISO exercise may also impact the AMT. For example, a qualified sale of ISOs might create a negative adjustment on form 6251, potentially leading to an AMT credit.
In a less direct way, a disqualified sale of ISOs generates taxable income. Your taxable income directly impacts the amount of room you have until you reach the AMT crossover point. If also could income AMT due and/or AMT credit, amongst other things.