When flying, you will need a government-issued ID card. Examples include your driver's license or state ID, passport, passport card or military ID. Having these documents will help avoid hassles when visiting various airport checkpoints.
If you are a noncitizen, your best bet is a passport. However, you may also use your permanent U.S. resident card or border crossing card. According to TSA.gov, foreign travelers are "not required to carry their passports if they have documents issued by the U.S. government such as Permanent Resident Cards."
You will need your ID at various points throughout the airport, including when you pick up your ticket and drop off any baggage, when you enter and exit security, when you board an aircraft and when you pick up your baggage at your destination. Having your ID ready will greatly reduce travel time.