Neutrino-electron scattering is a purely leptonic fundamental interaction and therefore provides an important channel to test the Standard Model, especially at the low energy-momentum transfer regime. We derived constraints on neutrino nonstardard interaction couplings depending on model-independent approaches which are described by a four-Fermi pointlike interaction and the unparticle physics model with tensorial components. Data on (nu) over bar (e) - e and nu(e) - e scattering from the TEXONO and LSND experiments, respectively, are used. The upper limits and the allowed regions of scalar, pseudoscalar, and tensorial nonstandard interaction couplings of neutrinos are derived at 90% confidence level in both one-parameter and two-parameter analysis. New upper limits for tensorial unparticle physics coupling constants and mass parameters are also placed.