Symmetry has a fairly new article on sterile neutrinos, explaining some of the basic ideas about what they are and why we’re interested in them. Measurements of the Z peak in e+/e- collisions at LEP showed that there are only 3 neutrinos, but there are some caveats to that. The measurements showing that there are 3 neutrinos really mean that there are 3 neutrinos that (1) have less than half the Z mass (91.2 GeV) and (2) interact with Standard Model particles via the weak force.
If we can instead add some “neutrinos” that don’t interact through the weak force, then there’s room for more neutrinos. One of the main ways people search for them is to find problems with the standard picture of 3-neutrino oscillations. If there are sterile neutrinos that mix with the three usual flavors (electron, muon, and tau), then maybe we can find evidence of neutrinos oscillating into sterile neutrinos. That is, regular neutrinos seemingly disappearing altogether rather than changing from one type to another.