Geoprocessing tools have been evolving since before the days of modern GIS technology. In those days, when geographic analysts were trying to solve real-world problems, they would gather around a whiteboard (or probably a chalkboard back then) and create flowcharts and diagrams very similar to those created with ModelBuilder today. The tasks in these early flowcharts evolved into software and have become the core set of geoprocessing tools you see today. This was an iterative process lasting many years (and is still ongoing) that has refined the tools into the set of simple and elemental operators that are installed with ArcGIS.
There are literally hundreds of tools, and learning what tool to use when takes time. Given the number of tools, it's beyond the scope of any single document to explain them all. The purpose of this document is to guide you to various other documents that describe the purpose and use of various tools.
The fundamental tools
The first place you should start is with An introduction to the commonly used GIS tools. This topic provides an in-depth look at common tools and their uses.
One effective way to learn about tools is to simply browse through the documentation. In the help system, expand the Geoprocessing book and the Geoprocessing tool reference book. This book contains documentation for all tools. Every tool is documented the same way; there is a short summary, notes about the usage of the tool, and a section on the exact syntax of the tool. The documentation often contains illustrations and will have links to other relevant topics. The documentation for an individual tool is often referred to as the tool reference page.