Within the Elgg framework, context can be used to by your plugin's functions to determine if they should run or not. You will be registering callbacks to be executed when particular events are triggered. Sometimes the events are generic and you only want to run your callback when your plugin caused the event to be triggered. In that case, you can use the page's context.
You can explicitly set the context with set_context(). The context is a string and typically you set it to the name of your plugin. You can retrieve the context with the function get_context().
If you don't set it, Elgg tries to guess the context. If the page was called through the page handler, the context is set to the name of the handler which was set in register_page_handler() (this is the string after pg in the url). If the page wasn't called through the page handler, it uses the name of your plugin directory. If it cannot determine that, it returns main as the default context.
Sometimes a view will return different HTML depending on the context. A plugin can take advantage of that by setting the context before calling elgg_view() on the view and then setting the context back. This is frequently done with the search context.