How to understand the difference between class_eval() and instance_eval()?

As the documentation says, class_eval evaluates the string or block in the context of the Module or Class. So the following pieces of code are equivalent:

class String
  def lowercase
    self.downcase
  end
end

String.class_eval do
  def lowercase
    self.downcase
  end
end

In each case, the String class has been reopened and a new method defined. That method is available across all instances of the class, so:

"This Is Confusing".lowercase 
=> "this is confusing"
"The Smiths on Charlie's Bus".lowercase
=> "the smiths on charlie's bus"

class_eval has a number of advantages over simply reopening the class. Firstly, you can easily call it on a variable, and it’s clear what your intent is. Another advantage is that it will fail if the class doesn’t exist. So the example below will fail as Array is spelt incorrectly. If the class was simply reopened, it would succeed (and a new incorrect Aray class would be defined):

Aray.class_eval do
  include MyAmazingArrayExtensions
end

Finally class_eval can take a string, which can be useful if you’re doing something a little more nefarious…

instance_eval on the other hand evaluates code against a single object instance:

confusing = "This Is Confusing"
confusing.instance_eval do
  def lowercase
    self.downcase
  end
end   

confusing.lowercase
=> "this is confusing"
"The Smiths on Charlie's Bus".lowercase
NoMethodError: undefined method ‘lowercase’ for "The Smiths on Charlie's Bus":String

So with instance_eval, the method is only defined for that single instance of a string.

So why does instance_eval on a Class define class methods?

Just as "This Is Confusing" and "The Smiths on Charlie's Bus" are both String instances, Array, String, Hash and all other classes are themselves instances of Class. You can check this by calling #class on them:

"This Is Confusing".class
=> String

String.class
=> Class

So when we call instance_eval it does the same on a class as it would on any other object. If we use instance_eval to define a method on a class, it will define a method for just that instance of class, not all classes. We might call that method a class method, but it is just an instance method for that particular class.