Reason why “XOR” is a linear operation, but ordinary “addition” isn’t? I'm new in cryptography and try to read some articles in this field.
Many of these articles talk about non-linear S-boxes, and nothing more on what they mean by their non-linearity.

I have a simple question which I think will guide me through my problem:


why  is the XOR operation is linear, but ordinary adding (+) not‽
what is the definition of linearity?
They are both linear, but in different algebraic Groups.  Which is to say, xor is linear in any finite field of characteristic 2, while 'ordinary' addition is linear in the infinite field of the Real numbers.  Addition modulo $n$ (which is more cryptologically significant than addition over the Reals) is also a linear operation, but in the ring of integers $\mathbb{Z}_n$.

A linear function is simply one that can be accurately described by an algebraic equation (in some particular algebra or another) where none of the terms are of degree greater than 1. 

So the xor of two variables can be accurately described by a linear algebraic equation in a finite field of characteristic 2.  But there is no linear equation in the algebra of $\mathbb{Z}_n$ (for $n &gt; 2$) that can accurately describe the xor of two variables.  Conversely, the addition modulo $n$ ($n &gt; 2$) of two variables can be accurately described by a linear equation in $\mathbb{Z}_n$, but not by a linear equation in any finite field of characteristic 2.

If none of that made any sense to you, that's OK.  You just need to read up about Abstract Algebra, and Fields, Rings and Groups.  It's a fascinating and beautiful area of mathematics, and much of cryptography will make no sense without at least some understanding of it.