That's a really great question, thank you for asking.

I will begin, in the classic anarchist tradition, of pointing out that I do not have all of the answers - this always sounds like a bit of a cop-out, but don't worry, I'll answer your question directly in a moment, and it's really important to communicate this because it's super important: anarchism isn't some blueprint for a society that we follow by rote and dogmatically implement, but rather a base layer of ideas we can use. As per an anarchist FAQ

Anarchists have always been reticent about spelling out their vision of the future in too much detail for it would be contrary to anarchist principles to be dogmatic about the precise forms the new society must take. Free people will create their own alternative institutions in response to conditions specific to their area as well as their needs, desires and hopes and it would be presumptuous of us to attempt to set forth universal policies in advance.

Also, I'll point out that my objection is to representative democracy, i.e. the current system, where voters are mere passive spectators of occasional, staged, and highly rehearsed debates among candidates pre-selected by the corporate elite, who pay for campaign expenses. The public is expected to choose simply on the basis of political ads and news sound bites. Once the choice is made, cumbersome and ineffective recall procedures insure that elected representatives can act more or less as they (or rather, their wealthy sponsors) please. My objection is not with people representing the opinions of others, as long as representatives have very limited power, a limited mandate, and as long as that power can be withdrawn and representatives recalled in a quick and easy way.

Anyways, with that out of the way, I will approach your question by explaining one possible system out of infinitely many which are possible.

Anarchists believe in structuring things from a bottom-up approach, so let's start at the bottom - each neighborhood could have a participatory community which makes decisions for that neighborhood around practical, everyday decisions that directly affect shared living. For example, maintenance of shared areas, the use of community buildings/rooms, improvements to the neighborhood, etc.

Of course, neighborhoods do not exist in a vacuum - neighborhoods would want to work together to share resources and to collaborate to achieve greater goals, so confederations of neighborhoods could be formed. Each neighborhood could select one or more delegates to attend confederation meetings and speak on the behalf of the neighborhood, but not make decisions on behalf of them. Instead, each delegate would attend the confederation meeting, and meet with their neighborhood to bring them news of what was discussed at the delegation meeting. Then, in the neighborhood meetings, each neighborhood would come to their own decisions around what should happen at the confederation level, and a delegate - not necessarily the same person as before - would take those decisions, questions, concerns and discussion points back to the confederation, where either a consensus would be reached, or further discussion - which could again, be brought back to the neighborhood to share. In fact, rotating the role of delegate would be a really good idea, so that multiple people can get a better idea of the issues in the wider community and understanding of how the bottom-up power structure functions.

The cool thing about anarchism is that we can experiment on these ideas and try out different things in a very responsive, small scale way, since the lower levels of participatory communities are small, even bad ideas have very limited harm if/when implemented, and other communities can learn from the failures of others.

I hope this makes sense, and I hope you see how this would differ from representative democracy. The main difference is where the power lies - under anarchism, the power to make decisions always remains with the people as a whole, rather than being concentrated into the hands of a very select few.

If you are interested in learning more, have any doubts about problems with the ideas I have outlined, or some other approaches which could be taken, I would strongly encourage you to have a read of an Anarchist FAQ's section about what an anarchist society would look like - I can pretty much guarantee that any question you have is answered somewhere in there, it's nothing if not comprehensive!

not every direct democratic system is majority rule. there are lots of ways to make decisions! remember, we don't need perfection, just something better than the tyranny of the minority we live under now. we can improve from there.

anyone seeking to sabotage decision making in bad faith can be handled however a given commune has decided to handle antisocial behavior (rehab, restorative justice, offering therapy, etc).

To add to the very good comment you already received, I would also point you towards an anarchist FAQ's answer to the question, "Won't there be a danger of a "tyranny of the majority" under libertarian socialism?". I will share a few choice paragraphs from there, but I encourage you to check it out yourself as it goes into a lot of detail:

Participation and self-management is the only way that majorities can come to see the point of minority ideas and for seeing the importance of protecting minority freedoms. This means that any attempt to restrict participation in the name of minority rights actually enforces the herd mentality, undermining minority and individual freedom rather than protecting it.

In the current system, voters are mere passive spectators of occasional, staged, and highly rehearsed debates among candidates pre-selected by the corporate elite, who pay for campaign expenses. The public is expected to choose simply on the basis of political ads and news sound bites. Once the choice is made, cumbersome and ineffective recall procedures insure that elected representatives can act more or less as they (or rather, their wealthy sponsors) please.

By contrast, in a libertarian society decisions are made following public discussion in community assemblies open to all. After decisions have been reached, outvoted minorities -- even minorities of one -- still have ample opportunity to present reasoned and persuasive counter-arguments to try to change the decision. This process of debate, disagreement, challenge, and counter-challenge, which goes on even after the defeated minority has temporarily acquiesced in the decision of the majority, is virtually absent in the representative system, where "tyranny of the majority" is truly a problem. In addition, minorities can secede from an association if the decision reached by it are truly offensive to them.

Zapatista sounds great, but...

for a couple of years

Put some respect on your elders.