The null hypothesis for a $\chi^2$ analysis of a contingency table is that the rows and columns are independent of each other. Under this null model, the expected count is:
$$
\hat E_{ij}=\left(\frac{\sum_j n_{ij}}{N}\right)\left(\frac{\sum_i n_{ij}}{N}\right)N
$$
In English: The row sum divided by the total $N$ is the probability of finding an observation in that row; the column sum over the total is the probability of finding an observation in that column; and if the rows and columns are independent, the probability of finding an observation in that cell is just the product of the marginal probabilities, and the expected count for that cell is simply that proportion of the total $N$.
What seems to be less well-known is that there can be residuals computed when your data are compared to the null model. For example, the Pearson residual for a cell in a contingency table is:
$$
r_{ij}=\frac{O_{ij}-\hat{E}_{ij}}{\sqrt{\hat{E}_{ij}}}
$$
I gather these are the residuals you are referring to.
These residuals are useful in visualizing contingency table data. For example, they are often used to 'shade' the boxes in a mosaic plot to make it immediately apparent which cells are out of line with expectations, in which direction, and by how much. (You can see an example of this in the mosaic [2nd] plot I posted in my recent question on visualizing contingency table data; note the legend explaining the colors on the side of the plot.)
Having covered these topics, we can answer your question easily: The sign indicates whether the number of observations with those particular characteristics (i.e., row and column classifications) is 'too many' (positive) or 'too few' (negative) relative to what would have been expected under the null model. The larger the residual is, the further that cell count diverges from what would have been expected under the null model. There is no evaluative component to this; positive residuals aren't 'better' than negative residuals. Finally, I agree with @PeterFlom that it makes no sense to say that categorical variables are "negatively correlated".
