Double strike is a static ability that modifies the rules for the combat damage step. (See rule 310, “Combat Damage Step.”)
At the start of the combat damage step, if at least one attacking or blocking creature has double strike or first strike, creatures without double strike or first strike (see rule 502.2, “First Strike”) don’t assign combat damage. Instead of proceeding to end of combat, the phase gets a second combat damage step to handle the remaining creatures. In the second combat damage step, surviving attackers and blockers that didn’t assign combat damage in the first step, plus any creatures with double strike, assign their combat damage.
Removing double strike from a creature during the first combat damage step will stop it from assigning combat damage in the second combat damage step.
Giving double strike to a creature with first strike after it has already put first strike combat damage onto the stack in the first combat damage step will allow the creature to assign combat damage in the second combat damage step.
Multiple instances of double strike on the same creature are redundant.