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

This “underscoring” seems to occur a lot, and I was wondering if this was a requirement in the Python language, or merely a matter of convention?

Also, could someone name and explain which functions tend to have the underscores, and why (__init__, for instance)?

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”.