There are several websites that search for the best fare including kayak.com, travelocity.com, and expedia.com. If you are looking for the best fare on last minute travel, also try fly.com. Search each site individually using the same criteria, i.e. the same dates, the same city pair and the same flights so you can be absolutely sure you are getting the lowest rate in the market. Sometimes, after you have found the lowest fare for particular dates, check alternate dates to see if there is a lower fare a day or two prior or after the dates you choose.
Most of the discounted airfare sites have boxes to check if you are flexible in your dates and flexible in arrival and departure cities. If your travel plans are not urgent, give yourself at least 14 days prior to your departure date to look at fares. Unless there is a seat sale in your market, most airlines require at least two weeks advance purchase notice for the best fares. Most airlines have a bereavement policy in place so if your travel is due to a critical illness or a death in the family, call the airline directly to book. Be prepared to give hospital or funeral home addresses and phone numbers in order to complete the reservation at the reduced rate.