Anarchism

Here's a super short summary on those points:

1. Anarchism is about horizontal power structures, mutuality, and community.

• Horizontal power rejects the idea that a small number of people with very large amounts of power have a right/duty to rule over a much larger group of people. Systems of power, (meaning groups of people and organizations that have influence), should be distributed widely, not concentrated or centralized.
• Mutuality means that the relationships that societies form with each other, both internally and externally, ought to be roughly equal. There are no relationships that fundamentally privilege one person or group above another. The rules apply equally to everybody.
• Community means that the primary consideration to any decision or action is how it affects the group first and foremost. For instance, instead of asking whether or not a certain action would generate profit, or make some specific person happier, the community involved as a whole should get to consider how said action would affect all of them.

2. There are many examples of anarchistic societies both past and present. The Amish, Buddhist monasteries, Anabaptist communities, the Rojava autonomous region in Syria, the Zapatistas in Mexico. While none of these groups 100% adhere to fully anarchist principles, they share many of the same principles and structures. Most people wonder how a society could function with no central leadership or power. While there are many things that would have to change greatly to make that work, there isn't anything about anarchism that inherently makes it impossible. Modern open source projects like the Fediverse we are on right now, operate in a pseudo-anarchistic way.
3. There are many ways to argue against Statism and centralized power structures, what you might find compelling will depend largely on your other ethical and moral commitments. But I will say for me, I never heard any arguments that could justify the power of the state. All arguments that seemed to justify state power ultimately could be flipped to argue for other things that obviously are bad, like mob rule. I also found it incredibly interesting that when pressed, most people actually agreed with me that there was no convincing justification for state monopoly on power, but they still rejected anarchism for pragmatic reasons. They didn't think modern societies could operated like that effectively.

Some of the other commenters here have linked good resources, For me, understanding two key things caused me to move to my political views:

• Capitalism is inherently self-defeating and unethical, which moved me to Socialism.
• The state monopoly on power is unjustifiable, which moved me to Anarchism. The combination of those two conclusions firmly places me in the Anarcho-Socialist camp of political philosophy. The details of how that should look and operate on a practical level are still something I discuss and debate with folks, but we all agree on the general principles.