Start with living relatives and work your way into the past. Ask questions and get as much detail as possible: full names, maiden names, date of birth, date of death, date of marriage, date of divorce, names of children, and places they may have lived. Family bibles often have familial information recorded in them, and old personal notes or letters may be helpful as well.
If all you have is a photo of your family member, ask other family members if they recognize him or her. Look through old family photo albums and try to figure out who is who. If you aren't sure when a photo was taken, check the back of the photo for the date.