Why do some functions have underscores "__" before and after the function name?

From the Python PEP 8 -- Style Guide for Python Code:

Descriptive: Naming Styles

The following special forms using leading or trailing underscores are recognized (these can generally be combined with any case convention):

• _single_leading_underscore: weak "internal use" indicator. E.g. from M import * does not import objects whose name starts with an underscore.

• single_trailing_underscore_: used by convention to avoid conflicts with Python keyword, e.g.

  Tkinter.Toplevel(master, class_='ClassName')

• __double_leading_underscore: when naming a class attribute, invokes name mangling (inside class FooBar, __boo becomes _FooBar__boo; see below).

• __double_leading_and_trailing_underscore__: "magic" objects or attributes that live in user-controlled namespaces. E.g. __init__, __import__ or __file__. Never invent such names; only use them as documented.

Note that names with double leading and trailing underscores are essentially reserved for Python itself: "Never invent such names; only use them as documented".

The other respondents are correct in describing the double leading and trailing underscores as a naming convention for "special" or "magic" methods.

While you can call these methods directly ([10, 20].__len__() for example), the presence of the underscores is a hint that these methods are intended to be invoked indirectly (len([10, 20]) for example). Most python operators have an associated "magic" method (for example, a[x] is the usual way of invoking a.__getitem__(x)).

Names surrounded by double underscores are "special" to Python. They're listed in the Python Language Reference, section 3, "Data model".

Actually I use _ method names when I need to differ between parent and child class names. I've read some codes that used this way of creating parent-child classes. As an example I can provide this code:

class ThreadableMixin:
   def start_worker(self):
       threading.Thread(target=self.worker).start()

   def worker(self):
      try:
        self._worker()
    except tornado.web.HTTPError, e:
        self.set_status(e.status_code)
    except:
        logging.error("_worker problem", exc_info=True)
        self.set_status(500)
    tornado.ioloop.IOLoop.instance().add_callback(self.async_callback(self.results))

...

and the child that have a _worker method

class Handler(tornado.web.RequestHandler, ThreadableMixin):
   def _worker(self):
      self.res = self.render_string("template.html",
        title = _("Title"),
        data = self.application.db.query("select ... where object_id=%s", self.object_id)
    )

...