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 flowcharts evolved into software and have become the core set of geoprocessing tools. 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 you see today.
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.
Commonly used tools
Commonly used tools gives a short description of some of (you guessed it) the most commonly used tools. From this document, you can to navigate directly to the tool's help reference page as well as the tool's dialog box.
Fundamental tool concepts
Fundamental tool concepts contains a more in-depth look at common tools and their uses.
Quick reference guide
The ArcGIS Geoprocessing Commands Quick Reference Guide is a book in PDF format that contains a brief description of every tool, its command line syntax, and some tips on using the tool.
Toolbox and toolset help
For any toolbox, you can right-click the toolbox and click Help. The toolbox help page describes the purpose of the toolbox and has links to overviews for each of the toolsets within the toolbox.
To learn more about the contents of toolboxes, follow these links:
An overview of the 3D Analyst toolboxAn overview of the Analysis toolboxAn overview of the Cartography toolboxAn overview of the Conversion toolboxAn overview of the Coverage toolboxAn overview of the Data Management toolboxAn overview of the Geocoding toolboxAn overview of the Geostatistical Analyst toolboxAn overview of the Linear Referencing toolboxAn overview of the Multidimension toolboxAn overview of the Network Analyst toolboxAn overview of the Spatial Analyst toolboxAn overview of the Spatial Statistics toolboxAn overview of the Samples toolbox