Psyche represents all the positive attributes we generally associate with the mind. It contains memory (both storage and recall), learning capability, analytical and calculative ability, and acuity of perception. Someone with a strong Psyche would be able to learn quickly and retain what they'd learned, take tests well, hold a job crunching numbers for a living, and other things reserved for 'smart people'.
It also measures less tangible and measurable attributes, such as self-discipline, the ability to manage stressful situations, and steadiness under fire.
In addition to the obvious, Psyche governs mental interactions with other living entities; it represents a character's ability to model (and interact with) the mental states of other Psyche-having beings, yielding empathy. These qualities contribute to the attractiveness of the character's personality.