In most situations, within the United States, you can still travel without ID, if possible get a police report. In all cases you will need to arrive at the airport very early for your flight and will be subjected to additional security checks. Additional information is available at https://www.tsa.gov/travel/security-screening/identification.
Different rules apply for international travel. If you are a US Citizen, you must contact the Department of State, or local embassy/consulate to replace a lost or stolen passport. It is always advisable to keep a copy of your passport in a separate location to expedite replacement in case of loss or theft.