I'll admit that this is one of the more confusing things out there in the world of aviation, and I may not have it 100% right, and you could probably write volumes on the subject of approved and unapproved parts, but:
If a part is Not TSO or PMA I will need to provide all testing data in order to install it under a field approval, good luck. (May already be an STC out there, which means all the work has been done.)
if a part is TSO but not PMA for my aircraft type I will need to determine if it can be installed, If there's an STC then great! someone's done all the work already. I can install this part under a field approval and it will be easier than a non-TSO or PMA part because the TSO means that it meets minimum standards for parts of that nature, but I will still need to show that it meets the requirements for installation in that particular aircraft. (Replaced some old Cessna wind up clocks with Mitchell electric ones recently, TSO but not PMA, no STC, but still quick and painless)
If a part is TSO and PMA I may need an STC to install it if it's a major alteration, or it's a direct replacement part and all I need is a log entry. A PMA part will have a list of aircraft that it's eligible for, I can't install a part that's a PMA for a PA-28 on a C-172. But say I'm installing McFarlane Aviation's elevator bushings for a C-172 on my C-172, they're PMA for it so nothing more than a log entry needed.
I will say that a couple years ago we began replacing our incandescent landing light bulbs (25hr life expectancy and 6 amp draw) with LED Landing light bulbs (10,000hr life expectancy 1.5. amp draw!). The bulbs met the TSO at the time but weren't PMA and if I remember right the first couple we did under a field approval which was relatively painless, then someone secured an STC for it so we used that. This allowed us to unscrew the old bulb and screw the new one on... I've heard that those same bulbs are PMA now but I haven't done one lately.
I hope that this is one of those things that gets streamlined along with the upcoming Part 23 revamp. Mostly I hope that the Part 23 revamp is NOT going to end up like LSA.