Fear and peace. I personally would list those two as the biggest effects that nuclear weapons have thus far proven to have on the world.

The existence of nuclear weapons, and the subsequent nuclear arms race, scared the hell out of many on our planet - and for good reason. What began as a couple low-yield nukes detonated over two Japanese cities became an arsenal that could take out a few cities, then rapidly expanded to a massive stockpile held by the Americans and Soviets in particular that had the insane destructive potential to obliterate a huge portion of our planet's population if they were ever to be used in full-scale anger. Never before had such awesome destructive power been placed under the control of so few people, and there were plenty of issues of trust from the citizens of not just the USA and USSR, but all around the world, regarding who had the final say in whether those incredible weapons were utilized or not and under what conditions, and whether the controls in place to govern the use of nukes were substantial enough.

The fear the citizens of Earth had and have was, and still is, justified, and it has forced many nations who did not possess those weapons to pick sides that they wanted to align with for protective support. This was a huge part of the Cold War divide, where some nations sided with America, and others were taken in under the Soviet sphere of influence.

Unfortunately, you cannot "uninvent" something like a nuclear weapon, and it can be extremely difficult to prevent other nations who had/have the willpower and resources from building their own nukes for their own arsenals and for the geo-political bargaining power they provide (nations like China, Israel, France, South Africa, the UK, Pakistan, India, North Korea).

Because of the incredible power of nuclear weapons and the power having them in an arsenal provided a nation who possessed them, nukes became a substantial geo-political bargaining chip in not just the US/Soviet relationship. The Mutually Assured Destruction (MAD) protocols led to an understanding between those nuclear nations that if you nuke me, I will still be able to nuke you before I am destroyed completely, and then we will both be dead, so it was appreciated as being far better for the Powers That Be to work at controlling direct tensions between the superpowers through peaceful diplomacy rather than allowing things to escalate to the point of direct military conflict.

Of course, the USA and USSR in particular still fought significant proxy wars, using third party nations as the way to blow off steam and battle it out without direct confrontation with each other, and that proxy "release valve" served to prevent dramatic escalation that would result in direct conflict between the superpowers with nuclear weapons potentially being employed as an offensive or defensive measure. In that sense, nuclear weapons ushered in an era of peace amongst those powerful countries through the overt threat potential of mutual annihilation.