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I would like to calculate the correlation between two variables. The first variable is continous and represents a performance measure. The second variable is an integer in the range from one to nine and represents an emotional rating of a user.

Currently, I'm using Spearmen correlation because neither the the first nor the second variable is normally distributed.

Is this correct? Or what correlation measure should be used for an integer variable?

Although your emotional rating data are coded as integers, are you sure that a difference between scores of 1 and 2 means the same as a difference between scores of 8 and 9 (etc.)? If not, what you have for those data are considered ordinal data, not strictly integer data. There are many threads on this site for how to analyze ordinal data.

With only 9 values for that ordinal variable there will be many ties, so Spearman's test might not be the best choice. One form of Kendall's test is designed to deal with ties.

You also might want to consider displaying the mean and standard error of your continuous variable over each of the 9 values of the emotional rating score. That could provide a useful visual display that might help convince anyone who is skeptical of your correlation results.

Normality and choice of tests

As others have noted in comments on this question, any of the Pearson, Spearman or Kendall correlation statistics might be useful here. All these statistics can be calculated regardless of the distributions of the data.

One issue is whether the assumptions needed to calculate p-values for the statistics are met. For example, the standard test for Pearson correlation assumes a bivariate normal distribution. That method of testing is what "requires" normality, you can still calculate the statistic without normality. Exact calculations of p-values typically can't be made for the other measures if there are ties. Yet even if the assumptions for tests aren't met, resampling can provide p-values and confidence intervals that are based on the empirical distributions rather than on assumed functional forms. A permutation test might be a good choice.

Kendall and Spearman statistics are based solely on the ranks of values along the two scales among the data points. As for other nonparametric tests they require no assumptions about distributions of the values. The tradeoff is that tests of those statistics might have less ability to detect true associations than the Pearson test if the assumptions of the Pearson test held.

So: if you have a linear relation between your performance measure and your emotional rating (that is, each step along the emotional rating scale leads to the same change in performance) then Pearson would be the best correlation measure. You could also consider standard linear regression of performance against emotional rating, which might provide more information. Then any requirement for normality in statistical tests would with respect to residuals about the means for each of the emotional rating groups (weaker than the assumption of bivariate normality), and you would have ways to gauge whether the linearity assumption is met.

For choosing between Kendall and Spearman, Kendall is essentially based on absolute values of differences while Spearman is based on squares of differences. Thus Spearman may be more affected by outliers in the relative rankings.

Yes Spearman Rank Correlation is the correct approach here. It is a non-parametric approach (solving the non-normality you describe) and is just the Pearson Correlation between the rank values of two variables. In other words in converts the variables of interests values to ranks and applies Pearson Correlation to the two sets of ranks.