We propose a new measure of dependence between a categorical random  variable and a random vector with potentially high dimensions, named semi-distance correlation. It is an interesting extension of distance correlation proposed by Szekely, Rizzo and Bakirov (2007) to accommodate the information of the categorical random variable. It equals zero if and only if the categorical random variable and the other random vector are independent. Two important applications of semi-distance correlation are considered. First, we develop a semi-distance independence test between a categorical random variable and a random vector. Under the null hypothesis, we show that the test statistic converges in distribution to a weighted sum of the infinite number of independent chi-square random variables. Moreover, when the dimension of the random vector tends to infinity, we derive the explicit asymptotic normal distributions of the test statistic under both the null hypothesis and the alternative hypothesis, respectively, which allows us to compute p-values in an efficient and fast way for high dimensional data. Second, we propose to use the semi-distance correlation as a marginal utility between the response and a group of covariates to do groupwise variable screening for ultrahigh dimensional classification problems. The sure screening property has also been established. Monte Carlo simulations and a real data application are presented to demonstrate the excellent finite sample property of the proposed procedures. A new R package semidist is also developed to calculate the sample semi-distance correlation and implement the proposed independent test and variable screening procedures.