If you refer to statistical significance, it implies that the predictor is not good *enough*. In order to be statistically significant the predictor has to be of a size that is more unexpected under the assumption that there is no effect than a criterion that you decide on beforehand (alpha-level). The size of the effect you can detect depends on various factors, e.g. sample size. The unstandardized coefficient gives you the expected change in the dependent variable given a 1 unit change in the predictor. Note that everything depends on which other predictors you have in the model. A predictor may be substantially related to the dependent on its own, but non-significant once other predictors are included in the model. This may be because the predictors overlap. The standardized coefficents represent semi-partial correlations, which in turn represent the correlation between X (predictor) and Y (dependent) after controlling for the effect of other included variables on X. So, if X is still a good enough predictor it means that it can account for variance in Y that none of the other variables can account for.