[[ ... ]] needs white spaces ([ or [[ are commands)
(( ... )) with or without white spaces
([[ ... ]] and (( ... )) are BASH specific)
Logic
(( 0 )) : false / exit 1
(( 1 )) : true / exit 0
String
[[ $a == $b ]] or [[ $a = $b ]]
[[ $a == [0-9] : equal range 0-9
[[ $a != $b ]]
[[ $a != [0-9] : not equal range 0-9
[[ $a > $b ]]
[[ $a < $b ]]
[[ -v $a ]] : variable was set
[[ -z $a ]] : null / zero length
[[ -n $a ]] : not null
Integer
[[ $a -eq $b ]] or (( $a == $b )) ( but (( $a = $b )) set variable )
[[ $a -ne $b ]] or (( $a != $b ))
[[ $a -gt $b ]] or (( $a > $b ))
[[ $a -ge $b ]] or (( $a >= $b ))
[[ $a -lt $b ]] or (( $a < $b ))
[[ $a -le $b ]] or (( $a <= $b ))
File
[[ -e filedirlink ]] : exist
[[ -f file ]] : 'file' exist
[[ -d directory ]] : 'directory' exist
[[ -L link ]] : 'link' exist