The Chisquare test of independence (also called the Chisquared test) is a standard measure of association between two categorical variables. It determines whether there is a significant relationship between the variables. If the two categorical variables are independent of one another, knowing the value of one provides no information about the value of the other variable. If one depends on the other, it can be worthwhile to examine their relationship.
The crosstab does the Chisquare test for you, but it can be helpful to understand how the system derives the values in question.
We will use gender as our first categorical value and favorite colors as our second.

What is your favorite color?
 
Gender

Yellow

Green

Blue

Red

Orange

Black

Purple

Row Total

Female

137

320

754

369

74

159

449

2262

Male

59

343

1188

454

120

155

112

2431

Column
Total

196

663

1942

823

194

314

561

4693

Now, we compute the variable counts expected if the variables were independent. The row and column totals are used to calculate the expected counts for each Gender/Color combination. So, we multiply the row total by the column total, then divide that by the grand total.
For the Male/Orange combination, that's 2431 * 194, which is 471614, divided by 4693. Our expected value (rounded to the nearest whole number) is 100. I've put the expected value for each cell in parentheses and used red text to differentiate it.

What is your favorite color?  
Gender  Yellow  Green  Blue  Red  Orange  Black  Purple  Row Total 
Female  137 (94)  320 (320)  754 (936)  369 (397)  74 (94)  159 (151)  449 (270)  2262 
Male  59 (102)  343 (343)  1188 (1006)  454 (426)  120 (100)  155 (163)  112 (291)  2431 
Column Total  196  663  1942  823  194  314  561  4693 
Now, we will calculate the difference between the actual and expected values for every combination, square that difference, and divide the result by the expected value for that cell. Adding all of those values gives us the test statistic. Using our Male/Orange cell as an example, the actual value is 120. The expected value is 100. The difference is 20, and squaring 20 gives us 400. Dividing 400 by 100 gives us 4. I've placed that in curly brackets and used green text to differentiate it.

What is your favorite color?  
Gender  Yellow  Green  Blue  Red  Orange  Black  Purple  Row Total 
Female  137 (94){19.67}  320 (320) {0}  754 (936) {35.39}  369 (397) {1.97}  74 (94) {4.26}  159 (151) {0.42}  449 (270) {118.67}  2262 
Male  59 (102){18.13}  343 (343) {0}  1188 (1006) {32.93}  454 (426) {1.84}  120 (100){4}  155 (163) {0.39}  112 (291) {110.11}  2431 
Column Total  196  663  1942  823  194  314  561  4693 
To get our test statistic, we add all the values in green to get 347.78. That's our Χ^{2} statistic.
Next, we need to calculate our degrees of freedom (df), which depends on how many rows and columns we have. The formula is df = (r  1) * (c  1). We have two rows and seven columns, so df = 6.
We need our significance level, α. CivicScience uses a significance level of 0.05 for all crosstabs.
Now, we use a published Chisquare distribution table (PDF) to look up the Chisquare value for 6 degrees of freedom and a 0.05 significance level. The Chisquare value is 12.592.
Next, we compare the value of our test statistic (347.78) to the Chisquare value of 12.592. Because 347.78 > 12.592, we reject the hypothesis that gender and favorite colors are independent. We can conclude that the variables have a relationship, though we can't say what kind.
The same variables are shown in a crosstab. The percentage of difference from the expected value is shown in parentheses if it is significant. The text is red for values that are lower than expected and green for those that are higher.
The variables show a high strength of association, as stated in the summary line at the bottom.
In the summary line, you'll see the number of responses, the significance expressed by the Χ^{2} value, the df (degrees of freedom) value, the pvalue, and the strength of association, which uses the T coefficient.
Χ^{2} is the Chisquare statistic. This one differs slightly from what we calculated above due to rounding differences.
We also calculated our degrees of freedom (df) above.
The pvalue corresponds to the Chisquare statistic and represents the probability of no relationship between our categorical variables. The lower the pvalue, the more confident we can be that such a relationship exists. Results shown to be statistically significant have been adjusted according to the BenjaminiHochberg false discovery rate procedure.
The T coefficient is Tschuprow's (sometimes spelled Chuprov's) T. Tschuprow's T measures the strength of the relationship between the variables. Its value ranges from 0 to 1. The higher the value for Tschuprow's T, the stronger the relationship between the variables.