In addition to normal experience, you gain wexp for each of the weapon types you can use. Your wexp amount for Swords, for instance, would be seperate from your wexp for Lances.

You gain wexp every time you fight. For every battle you fight in, you gain 1 wexp in the category of the weapon you're using in the fight.

When you reach certain levels of wexp, just like when you get 100 xp and level up, your weapon levels will increase. Each time your wexp for a weapon type reaches a certain threshold, your wexp for that type resets to 0, and you gain the ability to use new weapons, as indicated by their weapon rank. Said thresholds are:

E -> D: 15 wexp
D -> C: 20 wexp
C -> B: 25 wexp
B -> A: 30 wexp
A -> S: 40 wexp
S -> SS: 60 wexp

Any weapon type that you have an S-rank in gives you a +5 bonus to Hit and Crit while you're using a weapon of that type; an SS-rank gives you an additional +5 bonus to Avoid and Dodge while using that type of weapon.