For an overview of all tools, see List of Tool Proficiencies.
Tool proficiencies measure your aptitude with certain tools and objects, and your overall skill, knowledge, and ability with these implements. As with skill proficiencies, a check is most often called upon by the DM, but may be requested by the PC if they feel they're knowledge associated with these implements can be used in a certain challenge.
Below is each tool proficiency listed, describing their use and giving an example of what one can use the tool for. Note that there are a multitude of options and possibilities not described here that still apply to each and every tool. When and how is done in conversation between DM and player.