A smoking rivet is a rivet that is working itself loose, or is already loose, that spews a trail of aluminum oxide on the surface skin. The rivet may have worked loose from excess vibration, improper installation, hardening, corrosion, or flexing of the airframe. Typically, you wouldn't be able to detect any movement in a smoking rivet if you pressed a finger on it, but it may be moving just enough under flight loads to permit aluminum oxide corrosion.
A smoking rivet may or may not be a problem warranting corrective action. It depends on where the rivet is located. A loose rivet on a wing rib or fuselage stringer may call for nothing more than observation, while a smoking rivet that secures a surface skin to a spar could be more serious. If you push down on a smoking rivet and the skin gives under the pressure, you should consult a mechanic immediately.
It can take time to determine if a suspected smoking rivet is the real thing or simply dirt, grease, or other grime. Corrosion treatment fluid sprayed on the in-side of wing structures usually weeps for months afterward out of every pore, including around rivet heads.
If a smoking rivet appears on a freshly painted airplane, watch it closely. If it can be washed clean and doesn't reappear, it probably was just dirt.