The Threequal operator

Apart from simple comparision with == or eql? there is a another one: Object#=== aka Threequal.
Its intention is to provide “Case Equality”. The Documention furthermore states:

For class Object, effectively the same as calling #==, but typically overridden by descendants to provide meaningful semantics in case statements.

It is meant to to be overwritten!
In some classes it is already achieved. For example Range#===:

(1..5) === 3 # => true

So there is a comparision going on between two kind of objects, a Range and a Fixnum. Briefly the Threequal operator technically is nothing more than a method expecting a parameter, similar to == or +.
But this one is special. The Threequal also is called internally when a case statement tries to compare. The Range class once again, evaluating the a certain body mass index (BMI):

case 24
  when 0..18
  when 18..25
    'Normal weight'
    'Adiposity (overweight)'
end # => 'Normal weight'

Please note, the Threequal has to be overwritten in the when-branches-object class, not in the one the case statement tries to compare.
So whenever semantic comparision is required overwriting Object#=== totally makes sense. Semantic comparision is emphasized.
Another scenario is teams having members. The Team class:

class Team
  def initialize *names                                                         
    @members = names                                                            

  def === name                                                                  
    @members.include? name                                                      

The team Threequal method verifies, if its members list contain the assigned name. Creating 2 teams:

sales = 'Alice', 'Bob'
development = 'Chris', 'Deborah'

Given there is no other option then using a case statement for verifying the team, Chris belongs to, it would probably look like:

case 'Chris'
  when sales then 'Sales'
  when development then 'Development'
  else 'None'
end # => "Development"

And it returns the right answer.