How to use double or single brackets, parentheses, curly braces

Question

I am confused by the usage of brackets, parentheses, curly braces in Bash, as well as the difference between their double or single forms. Is there a clear explanation?

Answer 1

Brackets

if [ CONDITION ]    Test construct  
if [[ CONDITION ]]  Extended test construct  
Array[1]=element1   Array initialization  
[a-z]               Range of characters within a Regular Expression
$[ expression ]     A non-standard & obsolete version of $(( expression )) [1]

[1] http://wiki.bash-hackers.org/scripting/obsolete

Curly Braces

${variable}                             Parameter substitution  
${!variable}                            Indirect variable reference  
{ command1; command2; . . . commandN; } Block of code  
{string1,string2,string3,...}           Brace expansion  
{a..z}                                  Extended brace expansion  
{}                                      Text replacement, after find and xargs

Parentheses

( command1; command2 )             Command group executed within a subshell  
Array=(element1 element2 element3) Array initialization  
result=$(COMMAND)                  Command substitution, new style  
>(COMMAND)                         Process substitution  
<(COMMAND)                         Process substitution 

Double Parentheses

(( var = 78 ))            Integer arithmetic   
var=$(( 20 + 5 ))         Integer arithmetic, with variable assignment   
(( var++ ))               C-style variable increment   
(( var-- ))               C-style variable decrement   
(( var0 = var1<98?9:21 )) C-style ternary operation