Difference between class variables and class instance variables?

A class variable (@@) is shared among the class and all of its descendants. A class instance variable (@) is not shared by the class's descendants.

Class variable (@@)

Let's have a class Foo with a class variable @@i, and accessors for reading and writing @@i:

class Foo

  @@i = 1

  def self.i
    @@i
  end

  def self.i=(value)
    @@i = value
  end

end

And a derived class:

class Bar < Foo
end

We see that Foo and Bar have the same value for @@i:

p Foo.i    # => 1
p Bar.i    # => 1

And changing @@i in one changes it in both:

Bar.i = 2
p Foo.i    # => 2
p Bar.i    # => 2

Class instance variable (@)

Let's make a simple class with a class instance variable @i and accessors for reading and writing @i:

class Foo

  @i = 1

  def self.i
    @i
  end

  def self.i=(value)
    @i = value
  end

end

And a derived class:

class Bar < Foo
end

We see that although Bar inherits the accessors for @i, it does not inherit @i itself:

p Foo.i    # => 1
p Bar.i    # => nil

We can set Bar's @i without affecting Foo's @i:

Bar.i = 2
p Foo.i    # => 1
p Bar.i    # => 2

First you must understand that classes are instances too -- instances of the Class class.

Once you understand that, you can understand that a class can have instance variables associated with it just as a regular (read: non-class) object can.

Hello = Class.new

# setting an instance variable on the Hello class
Hello.instance_variable_set(:@var, "good morning!")

# getting an instance variable on the Hello class
Hello.instance_variable_get(:@var) #=> "good morning!"

Note that an instance variable on Hello is completely unrelated to and distinct from an instance variable on an instance of Hello

hello = Hello.new

# setting an instance variable on an instance of Hello
hello.instance_variable_set(:@var, :"bad evening!")

# getting an instance variable on an instance of Hello
hello.instance_variable_get(:@var) #=> "bad evening!")

# see that it's distinct from @var on Hello
Hello.instance_variable_get(:@var) #=> "good morning!"

A class variable on the other hand is a kind of combination of the above two, as it accessible on Hello itself and its instances, as well as on subclasses of Hello and their instances:

HelloChild = Class.new(Hello)
Hello.class_variable_set(:@@class_var, "strange day!")
hello = Hello.new
hello_child = HelloChild.new

Hello.class_variable_get(:@@class_var) #=> "strange day!"
HelloChild.class_variable_get(:@@class_var) #=> "strange day!"
hello.singleton_class.class_variable_get(:@@class_var) #=> "strange day!"
hello_child.singleton_class.class_variable_get(:@@class_Var) #=> "strange day!"

Many people say to avoid class variables because of the strange behaviour above, and recommend the use of class instance variables instead.