A single star * unpacks a sequence or collection into positional arguments. Suppose we have

def add(a, b):
    return a + b

values = (1, 2)

Using the * unpacking operator, we can write s = add(*values), which will be equivalent to writing s = add(1, 2).

The double star ** does the same thing for a dictionary, providing values for named arguments:

values = { 'a': 1, 'b': 2 }
s = add(**values) # equivalent to add(a=1, b=2)

Both operators can be used for the same function call. For example, given:

def sum(a, b, c, d):
    return a + b + c + d

values1 = (1, 2)
values2 = { 'c': 10, 'd': 15 }

then s = add(*values1, **values2) is equivalent to s = sum(1, 2, c=10, d=15).

See also the relevant section of the tutorial in the Python documentation.

Similarly, * and ** can be used for parameters. Using * allows a function to accept any number of positional arguments, which will be collected into a single parameter:

def add(*values):
    s = 0
    for v in values:
        s = s + v
    return s

Now when the function is called like s = add(1, 2, 3, 4, 5), values will be the tuple (1, 2, 3, 4, 5) (which, of course, produces the result 15).

Similarly, a parameter marked with ** will receive a dict:

def get_a(**values):
    return values['a']

s = get_a(a=1, b=2)      # returns 1

this allows for specifying a large number of optional parameters without having to declare them.

Again, both can be combined:

def add(*values, **options):
    s = 0
    for i in values:
        s = s + i
    if "neg" in options:
        if options["neg"]:
            s = -s
    return s
        
s = add(1, 2, 3, 4, 5)            # returns 15
s = add(1, 2, 3, 4, 5, neg=True)  # returns -15
s = add(1, 2, 3, 4, 5, neg=False) # returns 15

In a function call, the single star turns a list into separate arguments (e.g. zip(*x) is the same as zip(x1, x2, x3) given x=[x1,x2,x3]) and the double star turns a dictionary into separate keyword arguments (e.g. f(**k) is the same as f(x=my_x, y=my_y) given k = {'x':my_x, 'y':my_y}.

In a function definition, it’s the other way around: the single star turns an arbitrary number of arguments into a list, and the double start turns an arbitrary number of keyword arguments into a dictionary. E.g. def foo(*x) means “foo takes an arbitrary number of arguments and they will be accessible through x (i.e. if the user calls foo(1,2,3), x will be (1, 2, 3))” and def bar(**k) means “bar takes an arbitrary number of keyword arguments and they will be accessible through k (i.e. if the user calls bar(x=42, y=23), k will be {'x': 42, 'y': 23})”.

I find this particularly useful for storing arguments for a function call.

For example, suppose I have some unit tests for a function ‘add’:

def add(a, b):
    return a + b

tests = { (1,4):5, (0, 0):0, (-1, 3):3 }

for test, result in tests.items():
    print('test: adding', test, '==', result, '---', add(*test) == result)

There is no other way to call add, other than manually doing something like add(test[0], test[1]), which is ugly. Also, if there are a variable number of variables, the code could get pretty ugly with all the if-statements you would need.

Another place this is useful is for defining Factory objects (objects that create objects for you).
Suppose you have some class Factory, that makes Car objects and returns them.
You could make it so that myFactory.make_car('red', 'bmw', '335ix') creates Car('red', 'bmw', '335ix'), then returns it.

def make_car(*args):
    return Car(*args)

This is also useful when you want to call the constructor of a superclass.

It is called the extended call syntax. From the documentation:

If the syntax *expression appears in the function call, expression must evaluate to a sequence. Elements from this sequence are treated as if they were additional positional arguments; if there are positional arguments x1,…, xN, and expression evaluates to a sequence y1, …, yM, this is equivalent to a call with M+N positional arguments x1, …, xN, y1, …, yM.

and:

If the syntax **expression appears in the function call, expression must evaluate to a mapping, the contents of which are treated as additional keyword arguments. In the case of a keyword appearing in both expression and as an explicit keyword argument, a TypeError exception is raised.

It is also useful when defining a dictionary that may have many keys with the same value. For Example,

d = {'a':'r1', 'b':'r1','c':'r1','d':'r2'}

can be written,

d = {**dict.fromkeys(['a','b','c'],'r1'),'d':'r2'}