How can I satisfy a part testing contract?

To satisfy a contract you must activate the stage with the component to be tested while all the check marks are green. The check marks simply indicate that the conditions are right to start the test. Once you activate the stage you should be awarded the contract and that is instant. Recovery of the vehicle is not required for the contract, although you do recover funds.

For most part tests, activating the stage with the part when the exact conditions are met is enough to fulfill the conditions. However, in some cases the test has to be run by right-clicking on the part and choosing "run test".

This usually applies to parts which are not involved in staging (I had to do this with a small gear bay I had to test in orbit), but I also had this when I had to test the LV-N engine in orbit. The conditions were met, I activated the stage with the engine, and the contract wasn't fulfilled. Then I right-clicked the engine and picked "run test" and the contract was fulfilled.