I'm a relative novice, so some of this might not be quite right.
*** MINOR SPOILERS MAYBE ***
When you level up, you get to add a point to an attribute, so for me as Astrologer, I choose INT most often.
A weapon will have minimum values (numbers) for some attributes before you can use it effectively. If any are red, don't bother equipping it.
A weapon will also have a letter for one of more attributes, which is the scaling. If I remember correctly, they're like E,D,C,B,A,S from worst to best. The better it is, the more your weapon improves when you upgrade that particular attribute.
When you strengthen your weapon using stones, it will sometimes improve the scaling as well. There are regular stones with levels, something like 1-5, each type allowing you to upgrade your weapon three levels. So, stone [1] can get you +1, +2, +3. Stone [2] for +4, +5, +6. Somber smithing stones are for a whole other tier of weapons.