You would appear to have a count as the outcome variable - these days such an outcome is often modeled through Poisson regression or Negative Binomial Distribution regression - the latter  if the distribution is conditionally markedly positively skew  (variance greater than the mean). 

see The Analysis of Count Data: A Gentle Introduction to Poisson Regression and Its Alternatives http://www.tandfonline.com/doi/abs/10.1080/00223890802634175?journalCode=hjpa20

http://www.ats.ucla.edu/stat/spss/dae/poissonreg.htm

If you mean categorical (ie, nominal) variables by the tem of discrete outcome, you may conduct a χ2 (chi-square) test. See http://sphweb.bumc.bu.edu/otlt/MPH-Modules/BS/BS704_HypothesisTesting-ChiSquare/BS704_HypothesisTesting-ChiSquare2.html

If you are referring to the case of ordinal variables, it is appropriate to use the nonparametric tests such as the Mann-Whitney U Test or Wilcoxon Test to test for differences between two groups or within-group change. 

You will test for normality (normal distribution) based on by two indicators called skewness and kurtosis when you are using continuous variables.  When the continuous data are distributed normally, you may use parametric statistical tests (eg, t test, ANOVA, etc.). Discrete variables are "distribution-free" - that is, nonparametric statistical tests are appropriate for use.

You would appear to have a count as the outcome variable - these days such an outcome is often modeled through Poisson regression or Negative Binomial Distribution regression - the latter  if the distribution is conditionally markedly positively skew  (variance greater than the mean). 

see The Analysis of Count Data: A Gentle Introduction to Poisson Regression and Its Alternatives http://www.tandfonline.com/doi/abs/10.1080/00223890802634175?journalCode=hjpa20

http://www.ats.ucla.edu/stat/spss/dae/poissonreg.htm

Nominal- 2 groups- chi square & Mc Nemar test; > 2 groups-  chi square, cochrane Q test.

Ordinal- 2 groups- Man Whitney U & Wilcoxon sign test; > 2 groups- Kruskal Wallis, Freiedman test.

Thank you Kelvyn!

Thank you Lin!