I have a question about the order in which things are applied. Let's say I'm playing my level three abjuration wizard and a goblin takes a swing at me with his weapon. The DM says it hits me, so I cast shield as a reaction. What the DM has not yet told me is that it is a critical hit, so the goblin hits me anyway. I take the damage out of my arcane ward, but does the casting of shield recharge my ward? Was the shield spell cast before or after my ward took the damage? To elaborate on that example, let's say my arcane ward has five points left in it and the critical hit does six points of damage. Does the shield spell boost it from five to six points (since shield is a level 1 abjuration spell) before I subtract damage from the hit? Or does the hit deplete my ward, then inflict an additional one point of damage, and then the spell recharges the arcane ward for one point?

The shield triggers before damage, so it will charge ward before damage.

In general a reaction occurs after the trigger. However, the Shield spell is one of the few exceptions for although it states the trigger is on being hit, the effect (+5 to AC) is applied before the attack actually hits you.

So in this instance:

Goblin swings, shield goes up, goblin's weapon hits the shield, breaks through, hits you, deals damage.

So, yes, the shield is applied before any damage so the shield spell would restore 2 points of your Arcane Ward's missing hitpoints (it's double spell level). If your ward had 5 hit points, it would increase to 7. Then if shield could not prevent the attack, the damage is applied to the ward first, with any remainder going to you.

So: ward: 5 hp out of 6+int mod (presumably, +2 or +3 at least), goblin attack dmg: 6, shield spell restores 2 hp to ward to 7 hp. Damage reduces ward to 1 hp, and you take none.