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)).