Test Design:
The test case development activity is started once the test planning activity is finished.Based on the requirement analysis, test conditions are written first and an approval is obtained from the QA lead.
This is the phase of STLC where testing team write down the detailed test cases. Once the test cases are ready then these test cases are reviewed by peer members or QA lead. Once the preparation of Test Case Development and Test Environment setup is completed then test execution phase can be kicked off.
Test Execution:
In this phase, testing team start executing test cases based on prepared test planning & prepared test cases in the prior step.
Once the test case is passed then same can be marked as Passed. If any test case is failed then corresponding defect can be reported to developer team via bug tracking system & bug can be linked for corresponding test case for further analysis. Ideally every failed test case should be associated with at least single bug.
Once the bug fixed by development team then same test case can be executed based on your testing planning.