These tips are mostly for classes without any other healing ability.
Its a close fight. Health is low and healing potions are not an option. You need healing or you're going to die. Your escape path is blocked. What do you do? stun/root/sheep or otherwise render your enemy unable to attack you and then bandage.
Beware, DoTs will interrupt a bandaging attempt, so don't use while under a DoT because it will still start the cooldown timer for bandaging and prevent you from using it if you need it soon after.
Although wasteful, with good timing (and practice) you can bandage even while being attacked. The few hundred precious hps from a tick or two can turn the tide of a very close battle.
Several bandages vendor for more than the cloth used to make them, so rather than vendor the cloth, make bandages and sell them instead.
Patch 1.7.0 (patch date:2005-09-22): Effects that make players immune to physical will no longer be immune to the "Recently Bandaged" effect from First Aid.