Contexts are simply a way for Aegir to store certain pieces of information that it thinks may be useful later. They are named so that they can be referred to easily.
Every server, platform and site in Aegir has an associated context. Each one of these contexts stores information about the associated entity. So, for example, a site context stores the base URI of the site. Later, the code responsible for informing the web server of the site's existence can look at this context and determine the requested URI.
In theory contexts allow most of the back-end commands in Aegir to operate just on the name of the context. This is an important concept: the alternative would be to try to guess everything that the command being invoked might want to know and pass it in as a series of command line arguments.
For example, the command that Aegir uses to clone a site just needs to be given the context of the site to clone and the platform to clone it to. From there the code can work out where the destination platform actually is, what servers it needs to talk to, and with what credentials.
Note that Drush also has the concept of a context, but this is not to be confused with Aegir contexts which are different. Aegir contexts are closer to, and stored as, Drush 'aliases'.