What I was taught was that there are several, general, types of non-market economies.

Sharing economies work though voluntarism. People do whatever they think to be useful and share the proceeds to whatever extent they can. If no one thinks something needs to be done, it isn't done. If a lot of people think something needs to be done, it's done frequently. I was told that this was the most common economic system in the ancient world. The Indus Valley Civilization, which occupied a significant portion of the Indian subcontinent, is believed to have used this system. The modern Free Software Foundation, which is quite large, and involves some very labor intensive projects, works like this.

Gift economies work through reciprocity. (Usually) exotic goods are gifted from one community to another with no expectation that the receivers will give anything back immediately (or even that that particular community will ever give anything directly back). While it might sound odd to us, it worked quite well and is believed to be the basis for wide-scale trade though-out the continents. A well-documented example, which I unfortunately can't remember the name of right now, existed in a large archipelago, where each society's unusual goods were gifted in a clockwise direction throughout the islands, resulting in the widespread distribution of goods that were produced only on one particular island. Currently, the Pay it Forward movement is practicing a gift economy.

Redistributive economies collect excess or exotic goods and then require a re-distributor to pass them out to members of the community in accordance with some standard or other. The potlatches are one example of this. Progressive taxation which pays for social services and below-cost goods for the impoverished are a modern example of this kind of economy.

(I'm sure research has progressed since I attended school, so new categories may have been added, for instance, I wasn't told about the vertical archipelago, but this is what I can remember.)

There are many variations and hybrids of each general system, and each system, or hybrid system, can be used for any particular economic domain in a society, resulting in very flexible economics.

Despite our fears of exploitation, there's very little evidence that this occurred. Probably the biggest lesson we can learn from this is that, given a system which doesn't require us to take from each other, we probably can play nice and still be quite productive, despite what modern economists assume.