t-test
Click on "Analyze" → "Bivariate Relationships" and specify (2-level) nominal scale for X and continuous scale for Y to run the analysis.
From the red triangle in the upper left corner of the One-Way ANOVA report, select Means/ANOVA/Pooled Tests (if variances are assumed to be equal) or Individual Variances t-Tests (if variances are assumed to be unequal) to view the t-test report.
Paired Data T-Test
In the Analyze > Advanced Models > Matched Pairs platform, specify the two columns (continuous) you want to compare as Y.
For instructions on how to compare two groups in JMP, please refer to the attached materials (materials and sample data can be obtained from "Attachments" on the right).
Instructions are also provided for t-tests and paired t-tests.