We now think that there are some personality traits with which we are born, while most theorists think much of our personality is developed by our interactions with the people around us and the experiences we have. There is some speculation that even the traits we are born with exist as a consequence of the environment of the womb, which means, if correct, that even before birth, we are being influenced by people and experiences. Different theorists, of course, have different ideas about all of this, and I have included an enotes link for a good overview of these.
A behaviorist would say that our personalities are formed with positive and negative reinforcement of our behaviors as we grow. For example, a child who is very curious might be rewarded and encouraged, by parents and teachers, for his or her curiosity. This is a child whose curious personality will flourish. On the other hand, such curiosity might be discouraged or even punished at home or in school, and the child will learn fairly quickly to repress this trait, which, if repressed, might even die. Such a trait is then said to be "extinguished." Similarly, a talkative child might be encouraged or repressed, which will positively or negatively reinforce this trait. The behaviorist thus sees the environment, people and experiences, as being central to the formation of personality.
A Freudian will say that personality is formed by the satisfactory resolution of sexual stages of development, and this, too, is dependent on the environment. An example of this is when parents do not let a child move on from being in love with a parent to the full maturity of transferring that love to a suitable partner. The mother who flirts with her son and the father who encourages an adolescent daughter to sit on his knee too often is getting in the way of that transfer of affection, both behaviors a Freudian therapist would see as pathological. Good parenting means letting one's children move along through the stages without making them get "stuck" in those stages. Not everyone agrees, of course, that sexuality is the key to personality development, but Freud still has remarkable resonance.
As we are better and better able to peer into the workings of the brain, neurology is likely to play a much stronger role in the theories of personality development. What we do, how we behave then, is a question of how our neurons interact. The neurons connect along pathways, and it is helpful to think of these as ruts in a road. The more a particular pathway is used, the deeper the rut, and the more likely this is to be the more well-traveled path, just like Frost's poem, or like that shortcut across a wide lawn that everyone becomes drawn to once the path is clear. If this is the case, then people and experiences create those paths. Many negative experiences will cause anxiety or depression paths to be followed. Many positive experiences will conversely cause the "happy" paths to be followed. This is a self-perpetuating situation, since even our brains follow the paths of least resistance. How we are treated, whether we are abused or encouraged by those around us, the good or bad experiences we have, all result in pathways amongst our neurons, pathways that make us optimists or pessimists, empathetic or unfeeling, risk-taking or risk-averse, once again, people and experiences having a powerful affect upon our personalities.
There are other theories, of course, from Maslow, from Jung, from Adler, just to name a few. But all in all, personality theory does seem to rest almost exclusively on the interaction between what one is born with and the interaction of those inherent traits with the world one is born into.