Non-XP terminology (?), not quite synonymous with ProgrammerTest (a UnitTest measures a unit of software, without specifying why, while a ProgrammerTest assists programmers in development, without specifying how).

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(Note that although most of the text on this Wiki uses the term UnitTest, the canonical ExtremeProgramming name has been changed to ProgrammerTest.)
UnitTests are programs written to run in batches and test classes. Each typically sends a class a fixed message and verifies it returns the predicted answer.
UnitTests are a key component of software engineering, and the ExtremeProgramming discipline leverages them to permit easy code changes. Developers write tests for every class they produce. The tests are intended to test every aspect of the class that could conceivably not work. (Do test adding and removing, don't test accessors.)
Key here is to CodeUnitTestFirst.
UnitTests are all combined into a huge suite of tests, using e.g. KentBeck's TestingFramework. When developers go to release new code, they run all the unit tests, not just theirs, on the integration machine. The tests must run at 100%. If any test fails, they figure out why and fix the problem. The problem certainly resides in something they did ... since they know the tests ran at 100% the last time anything was released.
Of course, sometimes the tests let something slip through. When that happens, developers unconditionally enhance the unit tests so that that problem, and any similar one that comes to mind, won't happen again.
Regression testing is any type of software testing which seeks to uncover regression bugs. Regression bugs occur whenever software functionality that previously worked as desired, stops working or no longer works in the same way that was previously planned. Typically regression bugs occur as an unintended consequence of program changes.
Tandem has Unit Testing available
Kind of. Tandem doesn't really have objects in the sense this term implies.
You could write classes of programs, but I've never seen anyone do that.
And you could test it with unit testers, and test scripts.
However, you have to purchase ENCORE (Capture and Replay) to use it. I set it up in my first job.
I haven't seen anyone use it since.
So, when BA Merchants talks about unit testing, I'd guess they don't know what they are saying.
They don't even have ENABLE or ENLITEN.