Maturity is often used to describe the physiological development of an individual, gauging whether [he] has fully developed their bodies. In psychology, maturity refers to an individual's ability to appropriately cope with stressful situations, as well as how [he] responds to traumatic or difficult circumstances. This can also include the person's attitude and behavioral responses to [his] environment and surroundings, especially when conflict arises. Therefore, psychological immaturity refers to one's inability to appropriately cope with stressful situations, traumatic events, and difficult circumstances. Psychological immaturity also encompasses an individual's attitude and how [he] reacts to changes in [his] environment.
One such example would be if someone's requests are rejected: if that person keeps calm and reasonably addresses the conflict by either accepting it or negotiating with the person who rejected their requests, that shows psychological maturity and an appropriate development of [his] emotional responses. If that same individual were to become angry and irate, however, and begin to argue with the person or refuse to accept the response given, that generally displays psychological immaturity.
This differentiates between circumstances and perceptions, though, so the definition of what "maturity" is can differ between two people. Generally, however, psychological maturity is fundamentally defined as the advanced stage of one's logical reasoning and emotional control. Conversely, psychological immaturity would be the lack of development of such qualities.
Refer to the related links for more information about psychological maturity/immaturity.