When asked this question, most of us reply first with our gender and then with the roles we occupy. I might say I'm a woman, a psychologist, an artist, and then turn to my relationships -- a spouse, a friend, and so on. The more roles we have the more buffers we have against stress if something in one role goes awry. If I get disappointed regarding the sale of a painting, I can reflect on the acceptance of one of my journal articles. This is the accepted understanding of identity.
Some of our roles loom very large for us -- mother or spouse, for example -- and that can be limiting. If we have to endure the death of a child or a spouse, it's often hard to understand who we are now.
There is a very different response we could give to "Who are you?" One that doesn't depend on personal or professional roles at all. Who am I? I'm a person who cares about people, who enjoys a sense of humor, who values generosity and consciousness. When we define ourselves in this way, we're less vulnerable to negative circumstances, able to grow, and less defined by labels than by our more meaningful personal characteristics.