I have a Hidden Power TM that I got from... some... person. It doesn't matter where I got it. I was thinking about using it on my Abra. I looked up on the bulbapedia how hidden power works, and got a butt-load of math that I don't wish to perform. Is there some simple lamens terms way to determine what pokemon (or when) I should use this TM on? If it matters I'm currently playing on Diamond.
Check out the "testing hidden power" section from Bulbapedia:
So in Platinum and later Gen 4/5 games you could easily check the move, but in Diamond/Pearl your best bet is to make use of a Kecleon. It's still fairly time consuming to set this up, all for what's almost always a weak to middling power move.
Note that Hidden Power's max Power is 70 (min 30), and you can't control what power your pokemon will have, save for breeding/catching a new one with specific (painfully specific) IVs. It's extremely rare that Hidden Power is particularly useful except as a mid-level damage dealing move when your pokemon hasn't naturally learned enough mediocre moves to fill out their skill set.
I'd recommend just not using it at all if you don't have some devious strategy in mind like teaching a Water Hidden Power to a Fire pokemon or something. For competitive purposes it's a great way to teach a Special attacker a move Type they wouldn't otherwise get, but for the other 99% of us it's yet another mediocre skill.
Too add to the answer provided by @Ben Hidden power is excellent when it can be taught to pokemon that have tough first stage training issues. Good examples are Abra who only has teleport to start or possibly Magikarp who only has splash. This way you can train them easily without the fustrating stage of swapping them in the beginning of every fight and suffering half exp across two pokemon at best.