What does the caret (^) operator do?

It's a bitwise XOR (exclusive OR).

It evaluates to True if and only if its arguments differ (one is True, the other is False).

To demonstrate:

>>> 0^0
0
>>> 1^1
0
>>> 1^0
1
>>> 0^1
1

To explain one of your own examples:

>>> 8^3
11

Think about it this way:

1000  # 8 (binary)
0011  # 3 (binary)
----  # APPLY XOR ('vertically')
1011  # result = 11 (binary)

It invokes the __xor__() or __rxor__() method of the object as needed, which for integer types does a bitwise exclusive-or.

It's a bit-by-bit exclusive-or. Binary bitwise operators are documented in chapter 5 of the Python Language Reference.

Generally speaking, the symbol ^ is an infix version of the __xor__ or __rxor__ methods. Whatever data types are placed to the right and left of the symbol must implement this function in a compatible way. For integers, it is the common XOR operation, but for example there is not a built-in definition of the function for type float with type int:

In [12]: 3 ^ 4
Out[12]: 7

In [13]: 3.3 ^ 4
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-13-858cc886783d> in <module>()
----> 1 3.3 ^ 4

TypeError: unsupported operand type(s) for ^: 'float' and 'int'

One neat thing about Python is that you can override this behavior in a class of your own. For example, in some languages the ^ symbol means exponentiation. You could do that this way, just as one example:

class Foo(float):
    def __xor__(self, other):
        return self ** other

Then something like this will work, and now, for instances of Foo only, the ^ symbol will mean exponentiation.

In [16]: x = Foo(3)

In [17]: x
Out[17]: 3.0

In [18]: x ^ 4
Out[18]: 81.0