Since the 4.7 version, Drupal has free tagging included by default. Unfortunately this only allows for a shared tag set for a node, so that when multiple people tag a node, those tags go into a common list. Who tagged the node is not remembered, anyone can remove any tags and add new ones (given the permission).
Some people argued that this system should be called folksonomy, because at that time that was the hype of the day. Luckily the free tagging name was kept. Although if you need real folksonomy support (per user tagging, editing of own tags, personal tag clouds but still powering a common tag set), you need to look into a contributed module. Interestingly there are two similar solutions for the problem: the community_tags module and the taxonomy_user module. Since I need this functionality for a project, I needed to look at them, so here is a rundown of the differences I see from looking at their implementation.