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:
Players are able to test what the type of their Hidden Power is most often by utilizing a Kecleon. It does not have to be their own, though this makes it more convenient. Entering any Double Battle with the Pokémon with Hidden Power and Kecleon at the front of the list, then using Hidden Power on the Kecleon will cause it to change types to the type that Hidden Power is, or, if the Hidden Power is a Ghost-type move, cause nothing to happen due to Kecleon's immunity.
However, in Platinum, there is a man in the Veilstone Game Corner Prize Exchange house that will tell the player the type of their Pokémon's Hidden Power, and, even if the Pokémon doesn't know Hidden Power, the type that it would have. In HeartGold and SoulSilver, he is present in the Celadon Game Corner Prize Exchange house. In Black and White, he is present in the Pokémon Center in Mistralton City. In Black 2 and White 2, he's been relocated to the Pokémon World Tournament.
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.