Let's get this out of the way at the outset: on their own, I think psionics are pretty freaking' cool. Although almost universally ignored back in the day, psionics didn't fall by
the wayside because they were too byzantine or clumsy or whatever, they
withered on the vine because they seemed like you were playing a different game entirely. It was a fun game, but it just didn't fit with the rest of the D&D vibe.
Back when I was a kid, my gang's first batch of AD&D characters--all multi-classed elf munchkins with 19 Dexes and 18/72 Strengths--also miraculously succeeded at acquiring psionics even though elves are not on the short list of PC races that could be psionic (Humans, Dwarves, Halflings). We went around Psionic Blasting everything in town that we hadn't already Psychic Crushed. It was fun for a while but a few months later we graduated from munchkin gaming and psionics disappeared from our D&D experience ever after.
Enough of my self indulgence. Here's how to read the Psionics rules starting on pg 110 of the PHB:
The rules for acquiring psionix are actually pretty straight forward. You need an Int, Wis, or Cha (the psionic trinity) of 16 to have even a 1% chance of getting psionics. The author then lays out in analog form the possible additions to your roll, it woulda' been helpful to make a table like this:
|Even if all three are 16, that's still only a 1% chance.|
but he didn't. No big deal, at best, with 18s at I, W, and CH, you get a 9% chance of being psionically gifted. So even if you banish your best rolls to the classic dump stats, you still only had a slight chance of being rewarded.
e.g. Ponce the Paladin has a 17 Cha (minimum for paladins at the time), 15 Wis and 12 Int. Because one of those scores is 16 or higher, he gets to roll for psionics. He needs a 00 roll on d100, and rolls a 99. Close but no cig--"Wait, I get a 1/2% bonus because of my 17 Cha." he pleads. The Book says to ignore fractions, but we're all gamers here, we can handle fractions: roll a 10 sider, if you get 6 or higher you're in. Ponce rolls and gets another 9! Welcome to the Psionix Club, Ponce.
So you're one of the lucky few who managed the roll, or, more likely, your Dungeonmaster just said "Fine, you can have psionics." Now what? You have to determine your psionic strength; this is done by rolling d100 and adding 1 for each point of I, W, and/or CH exceeding 12. If two of those scores exceed 12, double the total bonus, if all three are 13 or higher, quadruple the crap out of the number.
Ponce rolls his 100-sider for psionic strength and gets a 25. To that number he adds (17-12=)5 points for his Ch, plus (15-12=)3 for Wis for a total of 8, doubled because 2 ability scores are >12 for a bonus of (2x8=) 16. 16+25 = 41.
Had Ponce's Int been 1 pt higher he'd have had a 9 base bonus, further, he would have been able to quadruple it for a total bonus of 36 to the d100 roll. Sadly, Ponce was not smart enough to be that smart so he's stuck with 41. This number is doubled to get total Psionic Strength, so Ponce has 82. It's called "psionic ability" in the PHB, but that strikes me as rather ambiguous, so I will be using Psionic Strength or Capacity henceforth.
What do you do with your psionic strength? You attack people with it, you defend yourself [and maybe your cronies] with it, or you use spell-like functions called "disciplines" that allow you to do a lot of really cool things for just a few points of psionic strength per round. Given that you're probably not encountering other psionic beings every day, more often than not you're just using those disciplines to do cool things like telepathy, teleportation, telempathic projection, telekinesis, or maybe one of the disciplines that don't start with "tele-".
But really what made Psionics cool was the combat, and I'm running out of room here so we'll get to that tomorrow in what will also be the 300th post here at DiceChuckles Entertainment.