How can I check whether multicolinearity exist between categorical variables or numerical and categorical variables? I did a linear regression with 10 variables, including categorical and numeric variables. But although my $R^2$ was 0.8 there were only 2 variables that were statistically significant. 
Am I correct to suspicious about multicolinearity? How can I measure multicolinearity between categorical variables? I know that the independent variables should affect the dependent variable. 
Another problem might be that my sample size is small (100) which might be why there are only two significant variables.  Is that right?
 A: Your question addresses quite a few things. Multicollinearity does not depend on the number of predictors in a regression model but on how much these predictors are correlated. For numerical variables I suggest you look at the correlation matrix between them; relationships between categorical variables could be assessed by means of Chi-square tests and combinations of numerical/categorical variables using the appropriate parametric and/or non-parametric tests.  It is very important that you check the VIF and tolerance levels after you run your regression model to identify which variables cause issues and whether it would be plausible to omit some of them from the model. If predictors are highly associated another idea is to run a factor analysis and run a regression after substituting observed with latent variables. For categorical variables it might also help to create dummy variables instead.
Regarding the second point you are making, you might in fact have lower power with a smaller sample to detect significant effects, however it might very well be that some independent variabels are just not good predictors of the dependent variable. You are not giving information on the entry method of predictors in the model but it is also likely that some of the 10 predictors would be good predictos in simple regession models but not after controlling for the remaining 9. Prior to examining the effect of different entry methods on the final model you select, I emphasize once again the importance of examining multicollinearity issues as they result in non- trustworthy estimates for the regression coefficients and their standard errors. 
