This is a very good question and one that musicians everywhere will have a different answer for.  I start to answer this for myself by asking myself the following questions:

1.  Can I play the left hand and the right hand separately from memory at both a slow and fast tempo (speed)?
2.  Can I play the piece hands together from memory at both a slow and fast tempo?
3.  Can I play the piece from multiple starting points from memory?
4.  Have I executed accurately my plan for articulations, dynamics, and phrasing in the piece?
5.  Do I make the melodies sing?
6.  Have I put myself into the music?

The last question is perhaps the most difficult to answer.  For me, it often comes about because the piece in question conjures up a setting, situation, story and/or a feeling and it is these things that I try to communicate through my playing.  Sometimes I play with my eyes shut because I feel like the distance between myself and the music is decreased.  In other words, I can feel more like I am part of the music when I shut my eyes and I am playing.  Of course, when you shut your eyes, you are giving up the benefits of visual memory which can be extraordinarily important when executing large leaps but at times I do this anyways, even when I am performing, because I can find myself in the music more easily.