Difference between [[ ]] AND [ ] or (( )) AND ( ) in Bash

((...)) is the shell's arithmetic construct. The operators you can use are documented in the manual: 6.5 Shell Arithmetic

(...) is a grouping construct that executes the contained commands in a subshell: 3.2.4.3 Grouping Commands

[...] is the "legacy" conditional construct. Documentation is at 6.4 Bash Conditional Expressions

[[...]] does everything that [...] does. The difference is that word splitting and glob expansion are not performed for variables inside [[...]] so quoting the variables is not so crucial. Additionally, [[ can do pattern matching with the == operator and regular expression matching with the =~ operator.

The reason [[ 10 > 9 ]] gives you an unexpected result is that the > operator inside [[...]] is for string comparison and the string "10" is "less than" the string "9".

I would have put this in a comment, but am not yet allowed to.

One difference I have noticed between [] and [[]], is that in the former it is possible to use multiple comparisons.

The same comparison throws a syntax error in [[]]

a=1
b=2

This works:

$  [ "$a" -gt 0 -a "$b" -gt "$a" ] && { echo '[b>a>0]'; }
[b>a>0]

This does not work:

$ [[ $a -gt 0 -a $b -gt $a ]] && { echo '[[b>a>0]]'; }
-bash: syntax error in conditional expression
-bash: syntax error near `-a'

I haven't really dug in to see why that is so, but when using multiple comparisons, I use [].