Emissivity is a dimensionless ratio representing a material's ability to emit energy by radiation, defined as the energy radiated by a material compared to that of a black body at the same temperature. To prove that emissivity is dimensionless through dimensional analysis, one must analyze the formula and show that both the numerator and denominator share the same dimensions. The discussion references Wikipedia for foundational knowledge on emissivity and dimensional analysis. Participants emphasize the need for a deeper understanding of the derivation of the emissivity formula. The conversation highlights the importance of proving the dimensionless nature of emissivity through equations involving mass, length, and time.