I'm not sure how to explain it in non-technical terms, but the problems of neutrinos and their mass is basically: One is not allowed to add mass-terms to the S

I'm not sure how to explain it in non-technical terms, but the problems of neutrinos and their mass is basically: One is not allowed to add mass-terms to the Standard model (that is a term in the expression that explains the particle physics we know of), because they break symmetries that we know hold (gauge symmetry). So in order to add a mass-term it must be generated by a dynamical field -- the Higgs field -- such that those symmetries are still respected. Hence, the Higgs is able to give the mass-terms to the fermions. However, the neutrinos are not like the other fermions. Fermions can have different chirality, but the neutrino only has one chirality. Due to the neutrinos being different, the Higgs field needs to be different in order to also generate mass-terms to the neutrinos (or another mechanism than the Higgs-mechanism must be used). Whatever that is, it is probable that it involves something that is beyond the Standard model. Might be a bit technical, hopefully someone else can make it more understandable.