Mine isn't much of a quote, it's more like a poem or a story framed in the wall of the house I grew up in.

There was an important job to be done and Everybody was sure that Somebody would do it.

Anybody could have done it, but Nobody did it.

Somebody got angry about that because it was Everybody’s job.

Everybody thought that Anybody could do it, but Nobody realized that Everybody wouldn’t do it.

It ended up that Everybody blamed Somebody when Nobody did what Anybody could have done.

• Charles R. Swindoll

The lesson it thought me was "responsibility and the use of one's initiative".

Growing up, my father would use it to explain that we didn't need our mother to remind us of our individual responsibilities,instead we need to have learnt what we ought to do and do it without being told.
So rather than going about asking who poured the water on the floor, you could clean it up to prevent an unsuspecting or unseeing passer-by from slipping over it and falling.

Growing up and still growing, I have learnt to take responsibility for my own actions and use my initiative appropriately. I have ofcourse not mastered the art, but it remains an invaluable lesson.