They are part of the Python syntax and unlike using single (') or double (") quotes, they can pretty much never be interchanged.

Square and rounded brackets can mean so many different things in different circumstances. Just to give an example, one may think that both the following are identical:

a = [1,2,3]
a = (1,2,3)

as a[0] gives 1 in both cases. However, the first one is creating a list whereas the second is a tuple. These are different data types and not knowing the distinction can lead to difficulties.

Above is just one example where square and rounded brackets differ but there are many, many others. For example, in an expression such as:

4 * ((12 + 6) / 9)

using square brackets would lead to a syntax error as Python would think you were trying to create a nested list:

4 * [[12 + 6] / 9]

So hopefully you can see from above, that the two types of brackets do completely different things in situations which seem identical. There is no real rule of thumb for when one type does what. In general, I guess that square brackets are used mainly for lists and indexing things whereas rounded brackets are for calculations (as you would in maths) and functions etc.

Hope this helps you out a bit!

Square brackets are lists while parentheses are tuples.

A list is mutable, meaning you can change its contents:

>>> x = [1,2]
>>> x.append(3)
>>> x
[1, 2, 3]

while tuples are not:

>>> x = (1,2)
>>> x
(1, 2)
>>> x.append(3)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
AttributeError: 'tuple' object has no attribute 'append'

The other main difference is that a tuple is hashable, meaning that you can use it as a key to a dictionary, among other things. For example:

>>> x = (1,2)
>>> y = [1,2]
>>> z = {}
>>> z[x] = 3
>>> z
{(1, 2): 3}
>>> z[y] = 4
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
TypeError: unhashable type: 'list'

Note that, as many people have pointed out, you can add tuples together. For example:

>>> x = (1,2)
>>> x += (3,)
>>> x
(1, 2, 3)

However, this does not mean tuples are mutable. In the example above, a new tuple is constructed by adding together the two tuples as arguments. The original tuple is not modified. To demonstrate this, consider the following:

>>> x = (1,2)
>>> y = x
>>> x += (3,)
>>> x
(1, 2, 3)
>>> y
(1, 2)

Whereas, if you were to construct this same example with a list, y would also be updated:

>>> x = [1, 2]
>>> y = x
>>> x += [3]
>>> x
[1, 2, 3]
>>> y
[1, 2, 3]

Square brackets: []

Lists and indexing/lookup/slicing

• Lists: [], [1, 2, 3], [i**2 for i in range(5)]
• Indexing: 'abc'[0] → 'a'
• Lookup: {0: 10}[0] → 10
• Slicing: 'abc'[:2] → 'ab'

Parentheses: () (AKA "round brackets")

Tuples, order of operations, generator expressions, function calls and other syntax.

• Tuples: (), (1, 2, 3)
  • Although tuples can be created without parentheses: t = 1, 2 → (1, 2)
• Order of operations: (n-1)**2
• Generator expressions: (i**2 for i in range(5))
• Function or method calls: print(), int(), range(5), '1 2'.split(' ')
  • with a generator expression: sum(i**2 for i in range(5))

Curly braces: {}

Dictionaries and sets, as well as in string formatting

• Dicts: {}, {0: 10}, {i: i**2 for i in range(5)}
• Sets: {0}, {i**2 for i in range(5)}
  • Except the empty set: set()
• In string formatting to indicate replacement fields:
  • F-strings: f'{foobar}'
  • Format strings: '{}'.format(foobar)

Regular expressions

All of these brackets are also used in regex. Basically, [] are used for character classes, () for grouping, and {} for repetition. For details, see The Regular Expressions FAQ.

Angle brackets: <>

Used when representing certain objects like functions, classes, and class instances if the class doesn't override __repr__(), for example:

>>> print
<built-in function print>
>>> zip
<class 'zip'>
>>> zip()
<zip object at 0x7f95df5a7340>

(Note that these aren't proper Unicode angle brackets, like ⟨⟩, but repurposed less-than and greater-than signs.)

Here's a flowchart:

Is it a list? Then use brackets.
Is it a dict? Then use braces.
Otherwise, you probably want parentheses.

Probably the thing that is giving you the most confusion is tuple vs. list. The two are similar and can be used interchangeably in many circumstances. But the difference is that you can't add or remove elements from a tuple once you've built it.