Like in the title.

I always give:

Arcade Gannon - Elijah's Advanced LAER+ and Riot Armour without any helmet, because it looks cool.

Craig Boone - Anti Material Rifle+ and Riot Armour without any helmet, because his default head gear is pretty cool.

Veronica - Two-Step Goodbye and Remnats Power Armour, whole.

Lily - Oh, Baby! I wish she could equip armour.

Raul Tejada - Ranger Sequioa with full Riot Armour.

Rose of Sharon Cassidy - Riot Armour without helmet and All-American

What do you equip your companions with?