Neuroticism is a characteristic that's considered by modern psychologists as one of the "Big Five" personality traits used to measure people's personalities.
Basically, neuroticism is defined as how much that person tends to experience negative emotional states. For example, someone who is particularly neurotic (that's the adjective), is more prone to feeling anxious, depressed, and high-strung than the average person. They might cry more frequently, and be more sensitive to criticism. Someone with low neuroticism score is usually more calm, thick-skinned, and easy-going.
In case you're interested, the other five are openness to new experiences, conscientiousness, extraversion, and agreeableness.
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Neuroticism is a personality trait characterized by tendencies to experience negative emotions like anxiety, depression, and irritability. People high in neuroticism may be more prone to stress and worry, and may have difficulty regulating their emotions. It is one of the Big Five personality traits commonly used to describe individual differences in personality.