Suppose I have a non-linear model, say probit/logit, how can I understand the significance of a coefficient as opposed to the significance of it's marginal effect? Say, I just need to know the importance of a variable, would the former suffice?
Let's assume that we have a single index model $$E[Y|X]=g(x'\beta),$$ where $g(.)$ is some nonlinear function. Unless there are interactions or polynomial terms, the index function coefficients' significance and sign will agree with various types of marginal effect and also the size and magnitude (relative to one) of the logistic coefficients in case of the logit. This will be the case as long as the function $g(.)$ is monotonic. Their magnitude, however, will not tell you very much about their substantive, as opposed to statistical, significance. It is sometimes possible to bound the marginal effects using the index function coefficients. For the logit model, ME$\le .25 \beta$, and for the probit the multiplier is $0.4$. This comes from the derivative of the expectation having a maximum value at $x'\beta=0$. These bounds are typically not sufficiently tight to be very informative.
Sorry I am not allowed to comment therefore writing here.
What do you mean by significance of the marginal effect of a coefficient? To me the coefficients are the marginal effects in case of a model with no interaction.
Normally I would get 2 outputs from the logit model: coefficients and their p-values. The p-values next to each coefficient indicate the probability that the coefficient is not zero (with low values we say that there is an effect), and the value of the coefficient shows you the magnitude of the effect (called effect size).
Here is a comprehensive general example http://www.ats.ucla.edu/stat/mult_pkg/faq/general/odds_ratio.htm