I'll try to break it down simply
Anarchism is basically a rejection of state and authoritarian power in favor of a far more self governed sort of system. Anarchy isn't a rejection of all rules, but more so a rejection of authoritarian rules and hierarchies when they aren't shown to be beneficial to a society at large. In an anarchist system, if a group wants to enact a law or a rule, it's their job to prove it useful and the job of the public to deem it necessary.
For example, when a kid asks their parents why they can't do something, like crossing the street without looking both ways, that's an example of anarchic thought. When dad says "Because you'll get hit by a car, just watch". that's an example of a beneficial authoritarian ruling.
Most anarchism is socialist in nature, and this is when we get into talks of what private property is. In a socialist system, private property is unethical, and what socialists mean by private property is land, resources, and other means of production (factories, offices, ect). The reason that private property is considered unethical is because it reinforces social hierarchies (ruling classes, working classes, ect.) and exploitation.
This is different from personal property, which really ought to be protected. Personal property is anything you use personally. Your home, your car, your computer, and so on.