Cloud's ultimate weapon is gained when you defeat Ultimate Weapon.
Tifa's ultimate weapon is gained when you go into Wall Market and Cid is in your group and when you go into that little booth where there is a machine gun on the roof, Cid fixes the computer and then you talk to the computer and get Tifa's ultimate weapon.
Aries's ultimate weapon is gained in the Temple of the Ancients.
Cait Sith's ultimate weapon is in Shinra HQ, on the floor with all the lockers, in one of the lockers. You can only get it in the second disc, when you are searching for the Mako Cannon.
Cid's ultimate weapon is gained when you go to rockettown and talk to the old man looking at the place where the rocket used to be, AFTER the rocket is launched.
Yuffie's ultimate weapon is in the sunken Gelnika ship, in the room next to the save point.
Vincent's ultimate weapon is recieved when you go to Lucecria's Cave(GO IN DISC 3) and go through the sequence and leave, then come back after a while, and if you don't get it, then come back later.
Barret's ultimate weapon is gained when you are at the end of Disc 2 and in the tunnels, it is in one of the treasure chests, so get every one. Note: You must have barret in your party.
Red XIII's ultimate weapon is given to you by Bugenhagen, when you see that he is sick, go exchange a party member for Red and talk to Bugenhagen.
The Lifestream moving towards Midgar to stop Meteor.