Atrocities are actions which violate the UN charter. The commission of an atrocity results in economic sanctions, meaning that all energy you would normally get through trade is eliminated. Some factions take deeper offense to atrocities, and may declare vendetta on you, even if they were formerly pact brothers.
After the required technology has been researched, you may call a Planetary Council, and vote to eliminate the law against atrocities. The personalities of the various faction leaders, and their current military situation, will determine if they vote in favor of this proposal. You can always count on Chairman Yang's vote for this measure. Once passed, you may then do whatever horrors you want, without fear of economic sanctions, including nerve stapling unruly drones. The planet does not object to nerve gas attacks and other atrocities once the law has been repealed.
Committing atrocities before repealing the U.N. Charter lowers your integrity rating.
Actions considered atrocities
Nerve stapling
Nerve Stapling serves as a means to instantly end drone riots at the specific base where the stapling occurred, and prevents further riots for 10 years. Such an action is condemned by the Charter, and will result in 10 years of economic sanctions. Stapled drones become less loyal, more vulnerable to probe team attacks, and more likely to outright rebel and join another faction. If you staple a base too frequently, you will cease to have any effect on your rebellious drones.
Planet buster
The usage of a planet buster, on a city or a unit, is the ultimate atrocity, and if the UN Charter is in effect it will result in every human faction declaring vendetta upon you, including your allies. The vast amount of ecological damage will almost certainty trigger global warming, and Planet will begin sending waves of native lifeforms at you.
Nerve gas
Nerve gas is an atrocity that increases your offensive attack by +50%. Usage of this gas on a base cuts its population in half, and will completely destroy a base with a population of 1. Each time you use it against a human faction, you get an additional 10 years economic sanction placed upon you by all other human factions, including those in a submissive pact that you have already conquered. You will also be unable to get the faction you are fighting to surrender to you, or even agree to a truce, no matter how overwhelming your forces are towards them.
Using nerve gas against alien factions will not bother the human factions at all. The aliens of course will be rather pissed.
Nerve gas can not be used against native lifeforms.
If you keep using nerve gas to destroy enemy bases, then this somehow causes increasing amounts of ecological damage to all of your bases.
Genetic warfare
Probe teams can utilize genetic warfare to instantly murder half the population of the victimized base. Military units occupying the base will suffer an immense amount of damage.
Obliterating a base
If you have one military unit inside a base you can destroy it, or go through the actions menu to find that option. Doing this to one of your own bases is less of an atrocity than destroying a base you just conquered. Why murdering your own people is less of a crime than murdering conquered people more likely to rise up against you and cause problems, is not explained.
You can eliminate a base you control without committing atrocities by various means. You can stop them from harvesting nutrients, thus starving all but one of them(base squares produce enough food for feeding one citizen), and then eliminate the last one by producing a colony pod to depopulate them entirely. Note that you building a colony pod does not eliminate a base with population 1 on the two lowest settings of the game, Citizen and Specialist.
You can also move a mind worm next to an unguarded base and then click the option to release it into the wild. When it moves into the empty base, it'll destroy it. This tactic can also be used against your vassals or allies, to weaken them without causing vendetta. Simply make certain the military units within are weaker than the mindworms you release directly next to their base, and their base will be obliterated regardless of its population's size.