This article is from the Final-Fantasy and other Square Soft Games FAQ, by firstname.lastname@example.org (Nick Zitzmann) with numerous contributions by others.
Here's how to find and obtain each manual:
Cloud: You can purchase 'OmniSlash' as a prize at the Gold Saucer's
Battle Square during Disc 2 or 3.
Barret: To get the 'Catastrophe', talk to a girl in one of the houses
near the train tracks after you succeed or fail at the North
Corel Huge Materia Quest in Disc 2.
Tifa: Go to Nibleheim and enter Tifa's house during Disc 2. With
her in your party, play the piano in her house using this
X, S, T, L1+T, L1+S, X, S, T, L1+X, O, X, S, X
Play it repeatedly and Tifa will find a letter from her mentor,
as well as the Final Heaven manual.
Aeris: To get the 'Great Gospel', after getting the buggy, pilot it
_into_ Costa Del Sol. You can now ride back to Junon. Exit,
and drive the Buggy back over the river fords to a cave.
Inside the cave is a sleeping man who will tell you how many
battles you've been through. If the last two numbers of the
battles are the same (22, 155, 888), he'll give you a lump of
Mythril. Return to the weaponsmith's house on the peninsula
near the Gold Saucer, and give him the Mythril, then go up the
stairs and open the small metal lid at the end of the walkway
for the Great Gospel Manual.
Red 13: You'll win his manual (Cosmo Memory) if you go to Nibleheim,
open the iron safe, and defeat the boss inside, "Lost Number,"
during any disc.
Yuffie: You can get her manual (All Creation) if you go to Wutai and
beat her father, Godo, on the fifth floor of the pagoda
northwest of the village.
Cait: (He doesn't have an Extreme Limit Break.)
Vincent: (Refer to the 'Ultimate Weapon' section below.)
Cid: While exploring the sunken Shinra plane, open the chests to
find his Extreme Limit manual, 'Highwind'.