This is an interesting question and may be informed by a bit of historical background. According to The Character Project of Wake Forest University, character study was a major function of psychological research in the early 1900s. A study on children's moral decisions by Hartshorne et al turned the tide against character study (against the existence of something called character) with results that suggested that since there seemed to be no consistency in moral decisions made by children, there was in fact no character component of personality. Recent research at Yale on infants' and toddlers' moral decisions has helped ignite the burning fire of revived interest in morality, ethics, virtue and character by finding that different toddlers display significantly consistent responses to the same situations; these Yale results antithetically contradict the Hartshorne results. So, historically, at one point, the answer to "What is character?" would have been, "Nothing: it doesn't exist." This historical perspective can now help us explain what current research says character in fact is.
To start the definition of character, character is defined as a component of personality. This draws forth the question "What is personality?" to which we'll divert so we can come back to character. Personality is defined as the overarching system compelling a person's thinking, motivation and behaviors. Interestingly, though the Latin root persona means "mask," masks were used in Greek and then Roman theater to reveal the identity of the character role being acted onstage (in massive amphitheaters) and not, as we normally think of masks (as in masked balls and "mask" psychology theory), as a means to hide the identity of an individual (Wikipedia on Princeton.edu). Personality is believed to be a composite of influences of environment and biological determinants: "As with all personality traits, dispositional attitudes develop through a combination of one's biology and environment" ("Haters gonna hate?" Laura Poppick on LiveScience).
Personality can be defined as a dynamic and organized set of characteristics possessed by a person that uniquely influences his or her cognitions, motivations, and behaviors in various situations. (Wikipedia on Princeton.edu)
Personality refers to individual differences in characteristic patterns of thinking, feeling and behaving. The study of personality focuses on two broad areas: One is understanding individual differences in particular personality characteristics .... The other is understanding how the various parts of a person come together as a whole. (American Psychological Association, adaptation from Encyclopedia of Psychology)
By the definition of personality above, character is a component of personality that may be shaped by both biological and environmental factors. Character, being related to the areas of personality theory study of motivation, behavior and cognition, is the measure of ethical, moral and virtuous (goodness) reasoning, motivation and behavior. Bear in mind that this definition sits on the cusp, if you will, of a new direction in psychology as pointed out by Paul Bloom and Karen Wynn of Yale's Infant Cognition Center whose joint research has pointed out that even six-month-old infants have what Bloom calls "naive morality" and can make distinction and decisions based on morality, virtue and ethics and that these distinctions and decisions are consistent across a population of infants and toddlers. Thus, character may be proven to be more like temperament than like personality. This raises the next question, "What is temperament?"
Temperament is defined as the qualities that are present at birth and have no primary relation to environmental influences [though in utero environment (environment experienced in the fetus state) may or may not be included in this definition]. The basic nine temperament traits recognized by psychological theory can be molded by influence, that being intentional influence, e.g., parents, and unintentional influence, e.g., hostile school environments, but cannot be altogether changed. As result, temperament markers are a good predictor of future life outcome. Similarly, barring physiological intervention (e.g., brain trauma, Alzheimer's), temperament traits are consistent throughout life and can be predicted themselves. This means that while temperament traits can change in quality over time, they will not change in type over time: they will remain consistent over time despite changes in quality due to education (improved) or aging (declining) etc.
Bearing in mind that character may later come to be defined as innate, the differences between personality, character and temperament are that personality and character are defined as character being a component of personality with both dependent on the combination of biological factors and environmental factors while temperament is defined as present at birth and consistent throughout life despite environmental factors, which cannot change but can only shape, or mold, the expression of temperament.
American Psychological Association, adaptation from Encyclopedia of Psychology