What does `def self.function` name mean?

Contrary to other languages, Ruby has no class methods, but it has singleton methods attached to a particular object.

cat = String.new("cat")
def cat.speak
    'miaow'
end
cat.speak #=> "miaow" 
cat.singleton_methods #=> ["speak"] 

def cat.speak creates a singleton method attached to the object cat.

When you write class A, it is equivalent to A = Class.new :

A = Class.new
def A.speak
    "I'm class A"
end
A.speak #=> "I'm class A" 
A.singleton_methods #=> ["speak"] 

def A.speak creates a singleton method attached to the object A. We call it a class method of class A.

When you write

class A
    def self.c_method
        'in A#c_method'
    end
end

you create an instance of Class(*). Inside the class definition, Ruby sets self to this new instance of Class, which has been assigned to the constant A. Thus def self.c_method is equivalent to def cat.speak, that is to say you define a singleton method attached to the object self, which is currently the class A.

Now the class A has two singleton methods, that we commonly call class methods.

A.singleton_methods
 => ["c_method", "speak"] 

(*) technically, in this case where A has already been created by A = Class.new, class A reopens the existing class. That’s why we have two singleton methods at the end. But in the usual case where it is the first definition of a class, it means Class.new.