What is associativity of operators and why is it important?

For operators, associativity means that when the same operator appears in a row, then which operator occurence we apply first. In the following, let Q be the operator

a Q b Q c

If Q is left associative, then it evaluates as

(a Q b) Q c

And if it is right associative, then it evaluates as

a Q (b Q c)

It's important, since it changes the meaning of an expression. Consider the division operator with integer arithmetic, which is left associative

4 / 2 / 3    <=>    (4 / 2) / 3    <=> 2 / 3     = 0

If it were right associative, it would evaluate to an undefined expression, since you would divide by zero

4 / 2 / 3    <=>    4 / (2 / 3)    <=> 4 / 0     = undefined

Simple!!

Left Associative means we evaluate our expression from left to right

Right Associative means we evaluate our expression from right to left 

We know *, /, and % have same precedence, but as per associativity, answer may change:

For eg: We have expression: 4 * 8 / 2 % 5

Left associative:   (4 * 8) / 2 % 5 ==> (32 / 2) % 5 ==> 16 % 5 ==> 1

Right associative:  4 * 8 /(2 % 5) ==>  4 * ( 8 / 2) ==> 4 * 4 ==> 16

it is the order of evaluate for operators of the same precedence. The LEFT TO RIGHT or RIGHT TO LEFT order matters. For

3 - 2 - 1

if it is LEFT to RIGHT, then it is

(3 - 2) - 1

and is 0. If it is RIGHT to LEFT, then it is

3 - (2 - 1)

and it is 2. In most languages, we say that the minus operator has a LEFT TO RIGHT associativity.

Update 2020:

The situation about 3 - 2 - 1 might seem trivial, if the claim is, "of course we do it from left to right". But in other cases, such as if done in Ruby or in NodeJS:

$ irb
2.6.3 :001 > 2 ** 3 ** 2
 => 512 

The ** is "to the power of" operator. The associativity is from right to left. And it is

 2 ** (3 ** 2)

which is 2 ** 9, i.e., 512, instead of

(2 ** 3) ** 2

which is 8 ** 2, i.e., 64.

There are three kinds of associativity:

The Associative property in mathematics

Order of Operations in programming languages

Associativity in CPU caches.

The Associative property in mathematics is a property of operators such as addition (+). This property allows you to rearrange parentheses without changing the value of a statement, i.e.:

(a + b) + c = a + (b + c)

In programming languages, the associativity (or fixity) of an operator is a property that determines how operators of the same precedence are grouped in the absence of parentheses; i.e. in what order each operator is evaluated. This can differ between programming languages.

In CPU caches, associativity is a method of optimizing performance.