Property of Division by vector for a field Serge Lang in "Linear Algebra" on page 2 says that 

The essential thing about a field is that it is a set of elements
  which can be added and multiplied, in such a way that additon and
  multiplication satisfy the ordinary rules of arithmetic, and in such a
  way that one can divide by non-zero elements.

But in his book "Calculus of several variables" he says that "it is meaningless to divide by a vector" and also I read online that division by vector is not defined, then how come the presence of a multiplicative inverse/division of an element can be essential property that a Field must satisfy?
 A: It seems to me that you are implicitly (and incorrectly) assuming that fields and vector spaces are the same thing. The first passage from Lang concerns fields, and the second passage concerns vector spaces. Why do you think there is a contradiction here?
Or, if you are instead asking "if having division is so important, why don't we also require it for vector spaces", there are two good answers:


*

*There is no definition of multiplying two vectors in a vector space (scalar multiplication takes in a vector and a scalar, not two vectors), so what would it mean to have a multiplicative inverse anyway?

*Vector spaces and fields capture different notions, each extremely important, but serving different purposes. Cars and airplanes are both very important, but it would be ridiculous to require cars to have wings just because they're so great for airplanes.
Regarding the first answer: there's a important distinction between "can be defined" and "comes with by definition". By definition, if $K$ is a field and $V$ is set, then choosing


*

*a "vector addition" operation, i.e. a way of defining a vector $v+w$ for any vectors $v,w$

*a "scalar multiplication" operation, i.e. a way of defining a vector $av$ for any vector $v$ and scalar $a\in K$


is what it means to make $V$ into a vector space over the field $K$ (the operations must also satisfy some properties).
Nowhere in this definition is there any way specified of multiplying two vectors, i.e. two elements of $V$. Now given any vector space $V$, I can, if I want to, define some crazy rule for inputting two elements of $V$ and outputting another element of $V$, and name that "multiplication of vectors". However, the reason that is not acceptable is that we want reserve the word "multiplication" for operations that satisfy certain properties, and my randomly chosen operation will in general not satisfy those properties. Thus, when people say that something in mathematics is "not defined", of course what they mean is "cannot be defined in a way that is useful, or has the properties we would want".
It also happens that many important mathematical objects that are vector spaces also come with a reasonable or natural way of defining the multiplication of the vectors. This is not a contradiction; the point here is that the concept "vector space" does not include a vector multiplication operation, even though it may be possible, and natural, to have one in some cases.
