This applies to weapons, photon arts, clothing and units.
To bring up the menu to equip weapons and photon arts with, press F5.
For clothing and units, press F6.
This window is for the weapon in particular. Clicking on a slot, or a weapon icon occupying a given slot will bring up the window on the right. It will display a number of weapons that can be equipped. Clicking one will cause it to occupy the slot you clicked earlier.
Be aware. You may notice that the weapons in the window on the right have numbers in brackets. This is comparing the attack power and elemental % of the currently equipped weapon. That is to say, the weapon that is currently being displayed on your character’s back, not the weapon whose icon you clicked on necessarily. This is not the case on the units screen, where it is comparing the unit you’re seeking to replace to those that could replace it.
To replace a given equipped Photon Art on a weapon, click on its icon (or on a blank slot if none are equipped). The window that pops up on the right will only show Photon Arts that can be equipped to this weapon. Click on one to replace it. As you can see,you can have multiple of the same Photon Art equipped on a chain.