In particle physics, a resonance is the peak located around a certain energy found in cross sections of scattering experiments. These peaks are usually associated with subatomic particle (elementary or not) and their excitations. The peak energy of the resonance corresponds to the mass m of the particle. The width of the resonance (Γ), also called the particle decay width, is related to the lifetime (τ) of the particle by the relation
To start with, please read this answer to understand what a resonance is, and come back here afterwards.
In summary, a "resonance" appears under the form of a peak in invariant mass distributions in case a particle is produced "on mass shell". The peak value is at the "pole" of the differential production cross section (i.e., the maximum value of the differential cross section - which may be infinite) , which usually take the form of a Breit-Wigner:
Clean experimental conditions are to be understood with respect to other types of colliders.
Example 1: An event with 2 muons in the CMS detector at the LHC (in a configuration with still moderate background conditions). Each beam crossing comes with 30 simultaneous gluon-gluon interactions, out of which one may (or may not, more often than not) be interesting. The interesting gluon gluon interaction also comes with its "underlying event" made of the debris of the colliding protons.