In quantum field theory, resonance is an unstable particle/bound state. It is characterized by a complex pole off the real line in the S-matrix (which happens to be analytic). A sharp resonance is a resonance with a sharp peak in the S-matrix (which corresponds to a long lifetime compared to the reciprocal of its mass) while a broad resonance is a resonance with a spread out peak (which corresponds to a short lifetime relative to the reciprocal of its mass). If a resonance is too broad, it might not be considered as a particle at all even if it has a complex pole (far from the real line).
If the resonance happens to be a "fundamental particle" (i.e. described by a field of its own), it shows up as a complex pole off the real line in the 2-point connected correlation function (i.e. the propagator).
See also