If $Y$ is my dependent variable having more than $2$ categories (so $Y$ is a non-metric/Categorical variable) and $X_1, X_2, X_3\cdots X_n$ are my independent variables which are metric in nature.
Now I want to find out which of the $X$ variables are significant. What test should I use. Some anyone guide to me a documentation for a test.
Note: I am aware of regression.I know if $Y$ would have been a metric variable I would have used regression and seen which of the $\beta$'s are significant.
There are plenty of methods for statistical classification, you can start with the Wikipedia article. In particular, if you are interested in regression models, Multinomial logistic regression - may help. The multinomial case is for $Y$ with more then two levels. If your dependent value $Y$ has only two levels, then the "classical" logistic regression will suffice. The last case is probably the most intuitive and closer to the "regular" regression models, as by simple transformation (logit), which is the natural logarithm of the odds of interest, you get the familiar linear model.