Table of Weapons
If you click on the above link, you can see a list with the available weapons for Albel, including their statistics. It also lists the weapons other allies can use as well!

Thanks to Cosmonaut for the table.

Here's some additional information on some of the 'special' weapons.

Crimson scourge Side-Quest:
You can get the crimson scourge whenever you return to Ellicoor II after disc 2, AFTER you've completed the Sphere 211 bit and battled the boss-battles at Styx. You'll see one of the most important scenes when it comes to Albel, and you'll get the sword of course! All you have to do is go to Airyglyph and go to the throne room to start the scene. Albel will be talking to the king, and later take the trial. This isn't Albel's strongest weapon however.

Ultimate Weapon:
His ultimate weapon is called Kirakuten Shell. It is dropped by Frenzied Biwig (Sphere 211, Floor 147). However, as all the ultimate weapons it has negative factors you'll have to remove with the Invention option in the game. You'll have to synthesize and/or refining the weapons so that they'll become powerfull and usefull in battle.

Negative Factors: Chance of Breaking 20%
Chance of Breaking 20%
Chance of Breaking 20%
Chance of Breaking 20%
Chance of Breaking 20%
ATK +1000

For more info on refining and synthesizing, please check Gamefaqs.