How can I create an array from the values of another array's key?

Newer versions of PHP allow using array_map() with a function expression instead of a function name:

$arr2 = array_map(function($person) {
    return $person['name'];
}, $arr1);

But if you are using a PHP < 5.3, it is much easier to use a simple loop, since array_map() would require to define a (probably global) function for this simple operation.

$arr2 = array();

foreach ($arr1 as $person) {
    $arr2[] = $person['name'];
}

// $arr2 now contains all names

This can be done in still more simple way by using array_column

$arr2= array_column($arr1, 'name');

print_r($arr2); //Array ( [0] => tom [1] => nick )

array_column is used to get the columns of a sub-array.

$array = array(0 => array('name' => 'tom', 'age' => 22), 1 => array('name' => 'nick', 'age' => 18));
foreach($array as $arr => $a){
    $names[] = $array[$arr]["name"];
}

print_r($names); //Array ( [0] => tom [1] => nick )