The Majorana nature of massive neutrinos will be crucially probed in the next-generation experiments of the neutrinoless double-beta (0 nu 2 beta) decay. The effective mass term of this process, < m >(ee), may be contaminated by new physics. So how to interpret a discovery or null result of the 0 nu 2 beta decay in the foreseeable future is highly nontrivial. In this paper we introduce a novel three-dimensional description of vertical bar < m >(ee)vertical bar, which allows us to see its sensitivity to the lightest neutrino mass and two Majorana phases in a transparent way. We take a look at to what extent the free parameters of vertical bar < m >(ee)vertical bar can be well constrained provided a signal of the 0 nu 2 beta decay is observed someday. To fully explore lepton number violation, all the six effective Majorana mass terms < m >(alpha beta) (for alpha, beta = e, mu, tau) are calculated and their lower bounds are illustrated with the two-dimensional contour figures. The effect of possible new physics on the 0 nu 2 beta decay is also discussed in a model-independent way. We find that the result of vertical bar < m >(ee)vertical bar in the normal (or inverted) neutrino mass ordering case modified by the new physics effect may somewhat mimic that in the inverted (or normal) mass ordering case in the standard three-flavor scheme. Hence a proper interpretation of a discovery or null result of the 0 nu 2 beta decay may demand extra information from some other measurements.