Cardsplain

First strike adds a damage step. If any attacking or blocking creature has or when combat damage begins, those creatures deal their damage first, in a step of their own. Then every other creature deals damage in the regular step. A creature killed in the first step misses the regular step.

Double strike deals damage in both steps, once with the first strikers and again with everyone else.

Dwalin, WeaponmasterIron Hills Blacksmith
Dwalin, Weaponmaster has first strike, and Iron Hills Blacksmith has double strike.

Damage stays on a creature until the end of the turn, so a hit in the first step still counts in the regular step.

Next, you choose which attacker Dwalin blocks. Its first strike saves it only when its first hit is lethal.