I don't believe any Native American societies had what we would now recognise as a police force before European colonisation, which did introduce semi-modern Western forms of policing- which was ever developing of course- and later saw forced legal impositions, restricting traditional law and justice systems. Nowadays, there exist tribe court systems and Native American police within and across reservations, and many nations are pioneering or involved in traditional justice system as an alternative, with the most famous being the use of peacemaking and restitution within the Navajo nation.

Pre-European contact, Native American groups varied considerably, across time and place. Some groups had more reliable food sources and were agricultural and pastoral, there existed permanent settlements and even cities, such as Cahokia, and some groups were semi-nomadic. From the Navajo to the Cherokee to the Tuscarora, each with unique belief and justice systems. Throughout time, alliances and wars were made, splits and splinter groups were created, new traditions, customs and communities formed.

So obviously I can't speak for every or the majority of pre-Columbian Native Americans. Every group had variations in how they carried out justice or protected members of the community, however a common feature, but with some noticeable exceptions (larger settlements, for example, at one point, some of the Huhugam people lived in villages of two thousand or more), is community and kinship policing. Living in small individual communities, usually as part of a larger one, ensured close relations to form, with family and relatives being highly important in most groups. Due to living usually in close proximity to one another, working with one another, spending time with one another, social policing was a key form of preventing wrongdoing and also punishing wrongdoing.

Social or communal policing uses shame, fear of embarrassment or ridicule, close ties and a sense of obligation to members of the community and an inability to simply escape or forget your wrongdoing to ensure any 'crime' is punished harshly through social shame, gossip and community memory and discussion. Correction within societies where community based policing often uses shaming as a tool, with all community members being made aware of the crime as a result.

This form of policing of behaviour is common throughout small communities, and exists to some extent in the modern world as well. However it was certainly not- or usually not- the only form of correction or policing of behaviour, being largely unorganised and unpoliced itself, although many groups throughout the world have used an institutional form of humiliation, public embarrassment or publicising of crimes to the community to police behaviour and actions, enforcing a rigorous social code.

To look at specific attempts to prevent and police wrongdoing: in Navajo tradition, the main form of policing was in the form of peacemaking and restitution. Peacemaking was guided by a 'peacemaker', with it taking the form of an argument, discussion, debate or lecture, airing the views and opinions of the victim, the offender and others involved, such as relatives. It was effective in resolving many cases of what would now described as civil disputes, such as land disputes, but also what would now be classed as a criminal matter, such as domestic violence or accusations of theft, with the aim being to allow the victim and offender to come to an understanding and agreement.

Where necessary, restitution (nályééh) was also a common practice, with the payment or giving up of goods or objects of value, either to replace what was lost or to otherwise make up for the wrongdoing. Unlike fining, for example, restitution is not punitive and provides a form of restorative justice, responsive to the individualised requirements and desires of the victim and the situations and circumstances of the wrongdoer. Restitution restored harmony and cohesion within the community.

Ensuring restitution and/or peacemaking occurred when necessary was the duty of the peacemaker, the victim and the wrongdoer. There was no distinction between disputes over what would now be classed as a criminal or civil matter. None had any investigative capacity. The need for investigation, a key part of the modern policing, was largely absent in smaller or close knit communities, where, almost always, the victim and the offender were very easily identified.

In rare cases, such as those of particularly horrible crimes, or in the case of a wrongdoer not providing restitution and repeating or continuing their offences, shunning would be used, detaching the offender from society, removing them from the cultural and social life and support of the community. The length of this banishment was almost always limited, with the hope being that the banished would return and therefore harmony could be restored once more.

Not to be glib, but you should probably specify the time period you're asking about, since many tribes did indeed have police forces sometime before 1995 (the scope of this sub).