A base is a premade private server which has its own unique style of code and features. While they are actually just servers the same as any other, we refer to them as bases as they contain the standard level of content and systems such as client-server connectivity, combat, trading, some skills and minigames etc.

Most servers began as these, then people added more content and renamed them to different things.
The most popular private servers you may have heard of such as SoulSplit and Ikov started as PI servers but have since been completely revamped to overcome the many many problems PI had.

Base name
Apollo (2011)
Hyperion
Project Insanity (PI)
Asteria
RuneSource