0. The idea

Let's stick to real vector spaces for a moment, in other words, vector spaces with scalars in $\mathbb{R}$.

Consider the space $\mathbb{R}^n$ with the operations of addition and scalar multiplication. A vector space ``over $\mathbb{R}$'' is intended to be any set with an addition-like operation and some scalar-multiplication-like operation obeying all the laws that $\mathbb{R}^n$ does, for all $n$.

The trouble is that the operations of $\mathbb{R}^n$ obey infinitely many laws, if you consider not only familiar ones such as $v + w = w + v$, but also messier ones such as $3v + (3w - u) = 3(v + w) - u$.

Some laws can be proved from others. People have analyzed the situation and found a list of basic laws from which all the other laws can be proved. The definition of a vector space simply lists the basic laws.

The same laws work fine for scalars in any field.