Some of the Pre-Columbian civilizations that inhabited the Americas we know today like the Aztec Empire or the Inca Empire were rulled by monarchs, tlatoanis in the Aztec Empire and sapa incas in the Inca Empire but, were there civilizations not rulled by a monarch?
I've read that the nahua state of Tlaxcala was rulled not by a monarch but by a council and so far this is the only civilization I've encountered that didn't have a monarch as its ruler.
Were there other states besides Tlaxcala that could be considered republics or democracies in today's terms?
The Haudenosaunee (the Iroquois Confederacy) would count here. The upper echelon of the political structure were two councils: the Grand Council that met at Onondaga and the Clan Mother Council that met at Gaustauyea. The Grand Council and its manner of operation is well documented. The Clan Mother Council and its operations are significantly more obscured.
In this post, I go over the Grand Council in detail, so I'll be brief here. In colonial times and earlier, The Grand Council was composed of representatives from the Five Nations that made up the Haudenosaunee and the rules governing the body are enshrined in the Gayanashagowa - the Great Law of Peace. According to the Gayanashagowa, the Council is composed of a minimum of 50 members. In practice though, it was only 49 as one seat remained ceremonially empty after the death of its original occupant. These seats were not divided equally among the Five Nations, but regardless of number of representatives each Nation received only one collective vote in the council. In addition to these 50 fixed seats, there were also an unlimited number of "Pine Tree Chiefs" - individuals who could be appointed to the council for distinguished service to their nation. They primarily served in an advisory role.
The 50 fixed seats served at the pleasure of the clan mothers. Each of these seats was held by a specific family lineage, and the matriarch of that lineage chose their representative and could recall them if they failed to represent their interests well. Clan mothers could be stripped of their position as well if they abused their power, and if things got particularly out of hand, the seat could be removed from that lineage and passed to another.
The clan mothers themselves were elected positions. Each longhouse chose a clan mother from among its adult women. Elder women were preferred but this was not a requirement. These clan mothers represented their clan's interests at local councils, but the fifty clan mothers who also held power over the seats on the Grand Council appear to have attend a confederacy-level council at Gaustauyea.
While the Grand Council focused its deliberations on external affairs (warfare, trade, etc.), the Clan Mother Council was primarily concerned with internal affairs (distribution of food and land rights being some of the few things we know they handled).
Unfortunately, the American Revolution shattered the Haudenosaunee and none of its successors managed to reform the Clan Mother Council. The Six Nations of Grand River may have attempted to do so, as its from them that we have partial list of the seats on the council, but with increased adoption of Euro-american traditions came a decrease in the political power of women in society.