The New Weapon System
In previous Final Fantasies, you had to find new and better weapons for each of your characters to improve, amongst other things, how many hit points you took off the enemy. Final Fantasy 8 employs a different system whereby you improve your weapon's stats by 'customizing' it. Depending on what you add to your weapon, say Squall's Gunblade, it'll make it stronger and more accurate. It'll also make your weapon look different, and cooler, in battle.

Weapon Upgrading

Weapons Shop Balamb Town Timber Derring City F.H. Town Blacksmith Dollet City Esthar Capital Man in Ragnarok

Let's say, for example, that you start the game with Squall's weapon, the Gunblade. To upgrade it, firstly you need to read a weapon magazine (monthly) so you can learn how to do it and what components are needed. The first upgrade for Squall is to a Revolver. For this to happen, you need 6 'Magic Stone's' and 2 screws. When you have these items go to a Weapon Shop (Junk Shop?) and they will upgrade it for a fee (the fee is determined by the level of upgrade).

Click here to view a list of all the weapon magazines:- If you don't have all the Weapons Monthlies, acquire Tonberry's 'familiar' menu and purchase missing magazines from Esthar.

Rinoa's weapon before (left) and after (right).

Finding and Gaining the Upgrade Parts
The items you need for upgrading your character's weapon are littered thoughout the world ('littered' being the best word; most of the items are junk). They can be gained from various monsters by either stealing or defeating them.

Make sure you have all these available to you:

1) Mug Ability (Bahamut, Diablo): This will allow you to steal items when you attack physically (very important).
2) Rare Item: Allows you to win items from monsters easily.
3) Change Card -> Item: So you can change cards into items.

Many thanks to Kevin Mulia.