A binary operation is said to be associative if for all elements
a, b and c we have 
. For
convenience let's drop the symbol for the operation and just write
(ab)c = a(bc). The associative property then allows us to speak of abc
without having to worry about whether we should find the answer to ab first
and then that answer "multiplied" by c rather than evaluate bc first and
then "multiply" a with that answer. Which ever way we process the expression
we end up with the same element of the set. Note though that it does not say
we can do the product in any order (i.e. ab and ba may not have the same
value).
Sets that have an associative binary operation are known as semigroups. In many practical applications of studying binary operations on sets it is not unusual to discover they are associative but it is something that cannot be assumed. Indeed, one commonly known operation, the cross product on three dimensional real vectors is not associative. We should also guard against thinking that associativity implies identitities and inverses exist.
So why is it we need associativity to solve 3x=11 in the reals? Well, the inverse of 3 is 1/3 and so 1/3(3x) =11/3 and now we use associativity to rewrite this as (3/3) x = 11/3 etc.