Double parenthesis with and without dollar

In the following, I use "returns" to indicate return values and "produces" to indicate "substitutes the resulting text."

$(...) means execute the command in the parens in a subshell and produces its stdout. Example:

  $ echo "The current date is $(date)"
  The current date is Mon Jul  6 14:27:59 PDT 2015

(...) means run the commands listed in the parens in a subshell. Example:

  $ a=1; (a=2; echo "inside: a=$a"); echo "outside: a=$a"
  inside: a=2
  outside: a=1

$((...)) means perform arithmetic and produce the result of the calculation. Example:
```
  $ a=$((2+3)); echo "a=$a"
  a=5
```
((...)) means perform arithmetic, possibly changing the values of shell variables, but don't produce its result. Example:
```
  $ ((a=2+3)); echo "a=$a"
  a=5
```
Note that the return value of the calculation is returned, so it can be used in while or if.
${...} means produce the value of the shell variable named in the braces. Example:
```
  $ echo ${SHELL}
  /bin/bash
```
{...} means execute the commands in the braces as a group. Example:
```
  $ false || { echo "We failed"; exit 1; }
  We failed
```

More generally what does the dollar sign stand for?

It means whatever it means in the given context.

Adding to the answer above:

$ VAR=2
$ if [ $VAR -eq 2 ]

> then
> echo 'yes'
> fi
yes

[[...]] offers extended functionality to single square brackets. Particularly, it is useful for =~ operator (used in regular expressions). Example:

$ VAR='some string'
$ if [[ $VAR =~ [a-z] ]]; then
> echo 'is alphabetic'
> fi
is alphabetic

Reference:

https://linuxconfig.org/bash-scripting-parenthesis-explained

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