I'm trying to understand super()

The reason we use super is so that child classes that may be using cooperative multiple inheritance will call the correct next parent class function in the Method Resolution Order (MRO).

In Python 3, we can call it like this:

class ChildB(Base):
    def __init__(self):
        super().__init__()

In Python 2, we were required to call super like this with the defining class's name and self, but we'll avoid this from now on because it's redundant, slower (due to the name lookups), and more verbose (so update your Python if you haven't already!):

        super(ChildB, self).__init__()

Without super, you are limited in your ability to use multiple inheritance because you hard-wire the next parent's call:

        Base.__init__(self) # Avoid this.

I further explain below.

"What difference is there actually in this code?:"

class ChildA(Base):
    def __init__(self):
        Base.__init__(self)

class ChildB(Base):
    def __init__(self):
        super().__init__()

The primary difference in this code is that in ChildB you get a layer of indirection in the __init__ with super, which uses the class in which it is defined to determine the next class's __init__ to look up in the MRO.

I illustrate this difference in an answer at the canonical question, How to use 'super' in Python?, which demonstrates dependency injection and cooperative multiple inheritance.

If Python didn't have super

Here's code that's actually closely equivalent to super (how it's implemented in C, minus some checking and fallback behavior, and translated to Python):

class ChildB(Base):
    def __init__(self):
        mro = type(self).mro()
        check_next = mro.index(ChildB) + 1 # next after *this* class.
        while check_next < len(mro):
            next_class = mro[check_next]
            if '__init__' in next_class.__dict__:
                next_class.__init__(self)
                break
            check_next += 1

Written a little more like native Python:

class ChildB(Base):
    def __init__(self):
        mro = type(self).mro()
        for next_class in mro[mro.index(ChildB) + 1:]: # slice to end
            if hasattr(next_class, '__init__'):
                next_class.__init__(self)
                break

If we didn't have the super object, we'd have to write this manual code everywhere (or recreate it!) to ensure that we call the proper next method in the Method Resolution Order!

How does super do this in Python 3 without being told explicitly which class and instance from the method it was called from?

It gets the calling stack frame, and finds the class (implicitly stored as a local free variable, __class__, making the calling function a closure over the class) and the first argument to that function, which should be the instance or class that informs it which Method Resolution Order (MRO) to use.

Since it requires that first argument for the MRO, using super with static methods is impossible as they do not have access to the MRO of the class from which they are called.

Criticisms of other answers:

super() lets you avoid referring to the base class explicitly, which can be nice. . But the main advantage comes with multiple inheritance, where all sorts of fun stuff can happen. See the standard docs on super if you haven't already.

It's rather hand-wavey and doesn't tell us much, but the point of super is not to avoid writing the parent class. The point is to ensure that the next method in line in the method resolution order (MRO) is called. This becomes important in multiple inheritance.

I'll explain here.

class Base(object):
    def __init__(self):
        print("Base init'ed")

class ChildA(Base):
    def __init__(self):
        print("ChildA init'ed")
        Base.__init__(self)

class ChildB(Base):
    def __init__(self):
        print("ChildB init'ed")
        super().__init__()

And let's create a dependency that we want to be called after the Child:

class UserDependency(Base):
    def __init__(self):
        print("UserDependency init'ed")
        super().__init__()

Now remember, ChildB uses super, ChildA does not:

class UserA(ChildA, UserDependency):
    def __init__(self):
        print("UserA init'ed")
        super().__init__()

class UserB(ChildB, UserDependency):
    def __init__(self):
        print("UserB init'ed")
        super().__init__()

And UserA does not call the UserDependency method:

>>> UserA()
UserA init'ed
ChildA init'ed
Base init'ed
<__main__.UserA object at 0x0000000003403BA8>

But UserB does in-fact call UserDependency because ChildB invokes super:

>>> UserB()
UserB init'ed
ChildB init'ed
UserDependency init'ed
Base init'ed
<__main__.UserB object at 0x0000000003403438>

Criticism for another answer

In no circumstance should you do the following, which another answer suggests, as you'll definitely get errors when you subclass ChildB:

super(self.__class__, self).__init__()  # DON'T DO THIS! EVER.

(That answer is not clever or particularly interesting, but in spite of direct criticism in the comments and over 17 downvotes, the answerer persisted in suggesting it until a kind editor fixed his problem.)

Explanation: Using self.__class__ as a substitute for explicitly passing the class by name in super() will lead to recursion. super lets us look up the next parent in the MRO (see the first section of this answer) for child classes. If we tell super we're in the child's method, it will then lookup the next method in line (probably this same one we are calling it from) resulting in recursion, causing either a logical failure (as in the answerer's example) or a RuntimeError when the maximum recursion depth is exceeded.

class Polygon(object):
    def __init__(self, id):
        self.id = id

class Rectangle(Polygon):
    def __init__(self, id, width, height):
        super(self.__class__, self).__init__(id)
        self.shape = (width, height)

class Square(Rectangle):
    pass

>>> Square('a', 10, 10)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "<stdin>", line 3, in __init__
TypeError: __init__() missing 2 required positional arguments: 'width' and 'height'

Python 3's new super() calling method with no arguments fortunately allows us to sidestep this issue.

The benefits of super() in single-inheritance are minimal -- mostly, you don't have to hard-code the name of the base class into every method that uses its parent methods.

However, it's almost impossible to use multiple-inheritance without super(). This includes common idioms like mixins, interfaces, abstract classes, etc. This extends to code that later extends yours. If somebody later wanted to write a class that extended Child and a mixin, their code would not work properly.