Some examples:

Traditional test:

foo="some thing"
# check if value of foo is not empty
if [ -n "$foo" ] ; then... 
if test -n "$foo" ; then... 

test and [ are commands like any others, so the variable is split into words unless it's in quotes.

[[ ... ]] is a (newer) special shell construct, which works a bit differently, the most obvious thing being that it doesn't word-split variables:

if [[ -n $foo ]] ; then... 

Some documentation on [ and [[ here.

Arithmetic test:

foo=12 bar=3
if (( $foo + $bar == 15 )) ; then ...  

All of the above act like normal commands, and if can take any command:

# grep returns true if it finds something
if grep pattern file ; then ...

Or we can use multiple commands. Wrapping a set of commands in ( ... ) runs them in subshell, which has a copy of the state of the shell. (Assume that test_something tests something only in it's working directory.)

# this will move to $somedir only for the duration of the subshell 
if ( cd $somedir ; test_something ) ; then ...

# while here, the rest of the script will see the new working
# directory, even after the test
if cd $somedir ; test_something ; then ...

Some examples:

Traditional test:

foo="some thing"
# check if value of foo is not empty
if [ -n "$foo" ] ; then... 
if test -n "$foo" ; then... 

test and [ are commands like any others, so the variable is split into words unless it's in quotes.

New-style test

[[ ... ]] is a (newer) special shell construct, which works a bit differently, the most obvious thing being that it doesn't word-split variables:

if [[ -n $foo ]] ; then... 

Some documentation on [ and [[ here.

Arithmetic test:

foo=12 bar=3
if (( $foo + $bar == 15 )) ; then ...  

"Normal" commands:

All of the above act like normal commands, and if can take any command:

# grep returns true if it finds something
if grep pattern file ; then ...

Multiple commands:

Or we can use multiple commands. Wrapping a set of commands in ( ... ) runs them in subshell, creating a temporary copy of the shell's state (working directory, variables). If we need to run some program temporarily in another directory:

# this will move to $somedir only for the duration of the subshell 
if ( cd $somedir ; some_test ) ; then ...

# while here, the rest of the script will see the new working
# directory, even after the test
if cd $somedir ; some_test ; then ...