What is the difference between a function, an unbound method and a bound method?

I'm asking this question because of a discussion on the comment thread of this answer. I'm 90% of the way to getting my head round it.

In [1]: class A(object):  # class named 'A'
   ...:     def f1(self): pass
   ...:
In [2]: a = A()  # an instance

f1 exists in three different forms:

In [3]: a.f1  # a bound method
Out[3]: <bound method a.f1 of <__main__.A object at 0x039BE870>>
In [4]: A.f1  # an unbound method
Out[4]: <unbound method A.f1>
In [5]: a.__dict__['f1']  # doesn't exist
KeyError: 'f1'
In [6]: A.__dict__['f1']  # a function
Out[6]: <function __main__.f1>

What is the difference between the bound method, unbound method and function objects, all of which are described by f1? How does one call these three objects? How can they be transformed into each other? The documentation on this stuff is quite hard to understand.

A function is created by the def statement, or by lambda. Under Python 2, when a function appears within the body of a class statement (or is passed to a type class construction call), it is transformed into an unbound method. (Python 3 doesn't have unbound methods; see below.) When a function is accessed on a class instance, it is transformed into a bound method, that automatically supplies the instance to the method as the first self parameter.

def f1(self):
    pass

Here f1 is a function.

class C(object):
    f1 = f1

Now C.f1 is an unbound method.

>>> C.f1
<unbound method C.f1>
>>> C.f1.im_func is f1
True

We can also use the type class constructor:

>>> C2 = type('C2', (object,), {'f1': f1})
>>> C2.f1
<unbound method C2.f1>

We can convert f1 to an unbound method manually:

>>> import types
>>> types.MethodType(f1, None, C)
<unbound method C.f1>

Unbound methods are bound by access on a class instance:

>>> C().f1
<bound method C.f1 of <__main__.C object at 0x2abeecf87250>>

Access is translated into calling through the descriptor protocol:

>>> C.f1.__get__(C(), C)
<bound method C.f1 of <__main__.C object at 0x2abeecf871d0>>

Combining these:

>>> types.MethodType(f1, None, C).__get__(C(), C)
<bound method C.f1 of <__main__.C object at 0x2abeecf87310>>

Or directly:

>>> types.MethodType(f1, C(), C)                
<bound method C.f1 of <__main__.C object at 0x2abeecf871d0>>

The main difference between a function and an unbound method is that the latter knows which class it is bound to; calling or binding an unbound method requires an instance of its class type:

>>> f1(None)
>>> C.f1(None)
TypeError: unbound method f1() must be called with C instance as first argument (got NoneType instance instead)
>>> class D(object): pass
>>> f1.__get__(D(), D)
<bound method D.f1 of <__main__.D object at 0x7f6c98cfe290>>
>>> C.f1.__get__(D(), D)
<unbound method C.f1>

Since the difference between a function and an unbound method is pretty minimal, Python 3 gets rid of the distinction; under Python 3 accessing a function on a class instance just gives you the function itself:

>>> C.f1
<function f1 at 0x7fdd06c4cd40>
>>> C.f1 is f1
True

In both Python 2 and Python 3, then, these three are equivalent:

f1(C())
C.f1(C())
C().f1()

Binding a function to an instance has the effect of fixing its first parameter (conventionally called self) to the instance. Thus the bound method C().f1 is equivalent to either of:

(lamdba *args, **kwargs: f1(C(), *args, **kwargs))
functools.partial(f1, C())

is quite hard to understand

Well, it is quite a hard topic, and it has to do with descriptors.

Lets start with function. Everything is clear here - you just call it, all supplied arguments are passed while executing it:

>>> f = A.__dict__['f1']
>>> f(1)
1

Regular TypeError is raised in case of any problem with number of parameters:

>>> f()
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
TypeError: f1() takes exactly 1 argument (0 given)

Now, methods. Methods are functions with a bit of spices. Descriptors come in game here. As described in Data Model, A.f1 and A().f1 are translated into A.__dict__['f1'].__get__(None, A) and type(a).__dict__['f1'].__get__(a, type(a)) respectively. And results of these __get__'s differ from the raw f1 function. These objects are wrappers around the original f1 and contain some additional logic.

In case of unbound method this logic includes a check whether first argument is an instance of A:

>>> f = A.f1
>>> f()
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
TypeError: unbound method f1() must be called with A instance as first argument (got nothing instead)
>>> f(1)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
TypeError: unbound method f1() must be called with A instance as first argument (got int instance instead) 

If this check succeeds, it executes original f1 with that instance as first argument:

>>> f(A())
<__main__.A object at 0x800f238d0>

Note, that im_self attribute is None:

>>> f.im_self is None
True

In case of bound method this logic immediately supplies original f1 with an instance of A it was created of (this instance is actually stored in im_self attribute):

>>> f = A().f1
>>> f.im_self
<__main__.A object at 0x800f23950>
>>> f()
<__main__.A object at 0x800f23950>

So, bound mean that underlying function is bound to some instance. unbound mean that it is still bound, but only to a class.