A feeling is a mental association and reaction to what a body experiences or to an emotion. A feeling is triggered by either the experience or the emotion. For sentient species, a feeling is an interpretation of either the experience or their emotional state. This interpretation is influenced by memory, personal experiences, and beliefs. Unlike emotions, which can be observed, a feeling can be hidden. (TNG: "Shades of Gray", "Descent")
The Talosians of Talos IV could read the feelings of their specimens. After reading Vina's feelings, as well as her thoughts and her dreams of what would be a perfect man, the Talosians picked Captain Christopher Pike. (TOS: "The Cage")
The Vulcans experienced their feelings differently than Humans did. (TOS: "Where No Man Has Gone Before")
In most Betazoids, their telepathic gifts developed at adolescence, however, occasionally a child was born with their telepathic abilities switched on. Most Betazoids born that way never led a normal life as the noise of other peoples' thoughts and feelings were overwhelming, incomprehensible, and painful, especially to a child. (TNG: "Tin Man")
The emotion chip created by Doctor Noonien Soong would give Data basic emotions and simple feelings. (TNG: "Brothers")