I'm not much of a romantic, so I would like to provide a more grounded and practical approach to understanding this kotowaza, or Japanese saying. I agree with Wintersweet's analysis that this saying refers to a sense of ephemera, but I do not think it literally means "awareness" (despite the similarity of the words' appearances) or have any built in sense of pity.

"mono no aware" - a deconstruction
  • mono - thing, or person.
  • no - possessive, marks that what precedes it is the owner of what follows it. or, that the following is a function of what precedes it.
  • a(wu) - to meet, or gather, or match.
  • ~are - marker of passivity for verbs, and in this form, a nominalizer

What this statement approximates, then, is the English phrase "the way things come together", with the inflection of passivity to imply that nothing can be done to help or hinder its progress.

Some colloquialisms expressing similar sentiment: (Thank you, Shro0m and mkb!)

This writeup is an exercise in demystifying Japanese as a foreign language: Yes, the words do have a concrete meaning and are not understandable only through 8 disparate English concepts. If you have any other colloquialisms to add, please msg me.