What is the meaning of the negative numbers?

I am having troubles understanding what is the "meaning" or best way to think of the negative numbers. I am not really sure where is my confusion, so I would set some examples that make me trouble. Sorry for my informal language.

1. When thinking of "5 - 3" should I think about it as "taking away three elements from five elements" or is it "five positive elements and three negative elements together"? I mean, are negative numbers really numbers (or some quantity) or are they just a natural number under a subtraction operation.

2. From there come I think the rest of my questions. What does "10/-5" really mean? Am I dividing "10 elements" into "five negative groups"? How is that possible? How can there be negative groups?. How can the result be negative, since the elements I am dividing are positive?

I know how to get the results of these cases. But I do not really think I understand what a negative number really mean.

Can anyone help me see where is my confusion and if possible recomend some content to read?