Here's to elaborate on what everyone else said:
There are about 5 versions of the PSP:
PSP-100x -->The Phat (I have). 100% hackable. We use a Pandora's Battery and a Magic Memory Stick to hack these. When you install a custom firmware on these (aka 5.50 Gen D-3) it is Permanent.
Early model PSP-200x -->The Slim. 100% hackable. We also use a Pandora's Battery and a Magic Memory Stick to hack these. When you install a custom firmware on these (aka 5.50 Gen D-3) it is Permanent. It also has 2x the RAM (64MB compared to 32MB on the Slim) Best model in my opinion, but a bit hard to find.
PSP-200x v3 -->The Unhackable Slim. Referred to as v3 after the TA-088 v3 motherboard inside. These can be hacked, but with temporary custom firmwares (ChickHEN, 5.03 Gen, ala 3rd Gen iPod Touch models)
PSP-300x -->The "Brite." Contains the TA093 v3 motherboard (Same effects as the Unhackable slim), therefore can only take temporary custom firmwares.
PSP Go (I can't remember the series number off the top of my head) -->Completely unhackable as of right now. No UMD, No replaceable battery, No MemoryStick (M2 instead), this one is a NO Go.
Also, about the Slims that can be hacked (Quote The PSP FAQs Thread):
PSP-2000 Motherboard Information:
TA-85 (Can use and make Pandora Battery)
TA-85v2 (Can use, but cannot make Pandora Battery)
TA-88v2 (Can use, but cannot make Pandora Battery)
TA-88v3 (Cannot use or make Pandora Battery)
Limited Edition & New Boxstyle:
Daxter Limited Edition Pack (TA-85v2/3.71 OFW)(CONFIRMED)
God of War Limited Edition Pack (TA-88v2/3.95 OFW)(CONFIRMED)
Madden 09 Limited Edition Pack (TA-88v2/3.95 OFW & TA-88v3/4.01 OFW)(CONFIRMED)
Piano Black with the New Box Style (TA-88v3)(UNCONFIRMED)
Piano Black (Amazon.com)(TA-88v3)(PARTIALLY CONFIRMED)
PSP-2000 Original Boxstyle Box Codes (Thanks to Alek @ Dark-AleX.org):
(No Letter) = 3.60
A = 3.71
B = UNKNOWN
C = 3.72
D = UNKNOWN
E = 3.80
F = 3.90
G = 3.95 / 4.01 (TA-88v2 / TA-88v3) BUT
A PSP with G and the 4.01 firmware is most likely a TA-88v3 motherboard, so it is advised that you stay away from them.
Hope that helps! Also, the list of Limited Editions missed one: The StarWars Darth Vader model, that one IS 100% hackable.