I want to test whether the mean in two samples are different. I don't know their variance and estimate them from the sample. When should I use the Welch-Satterthwaite approximation to the degrees of freedom for the test statistic?
Welch's t-test does not assume equal variances for the two groups.
Some people argue that Welch's t-test should be used by default. For an example, see the article at this link.
Also note that R uses Welch's t-test as the default for the t.test function. I believe SAS output for PROC TTEST returns both the Student t, and Welch's t ("Satterthwaite"). See Wikipedia for other software implementations.
For a more complete discussion, see this thread on Cross Validated