wp_enqueue_script() │ WP 2.1.0

For example if you want to replace the default jquery script with a CDN-version, add this code to the functions.php:

add_action( 'wp_enqueue_scripts', 'my_scripts_method', 11 );
function my_scripts_method() {
	wp_deregister_script( 'jquery' );
	wp_register_script( 'jquery', 'http://ajax.googleapis.com/ajax/libs/jquery/1.7.2/jquery.min.js' );
	wp_enqueue_script( 'jquery' );
}    

// For frontend of the site
add_action( 'wp_enqueue_scripts', 'my_scripts_method' ); 

function my_scripts_method() {
	wp_enqueue_script( 'scriptaculous' );            
}

Register and enqueue a custom script that depends on jquery (so it will be loaded before this script).

add_action( 'wp_enqueue_scripts', 'my_scripts_method' );
function my_scripts_method() {
	$script_url = plugins_url( '/js/newscript.js', __FILE__ );
	wp_enqueue_script('custom-script', $script_url, array('jquery') );
}

For example you need conditional tags for loading script scriptaculous. You can use hook wp because conditionals tags available after it.

add_action( 'wp', 'add_my_script_where_it_necessery' );
function add_my_script_where_it_necessery(){
	if( is_single() )
		add_action( 'wp_enqueue_scripts', 'my_scripts_method' );
}
function my_scripts_method() {
	$script_url = plugins_url( '/js/newscript.js', __FILE__ );
	wp_enqueue_script( 'newscript', $script_url, ['scriptaculous'] );
}

Specify the name of the dependency in the third argument.

Often it is required that another JavaScript file be loaded before the JavaScript files that come with the design theme. WordPress provides an API that allows you to specify its dependencies when registering a script. For example, the theme with the code below requires that the jQuery library be loaded before the custom_script.js script:

add_action( 'wp_enqueue_scripts', 'my_scripts_method' );

function my_scripts_method() {

	wp_enqueue_script( 'custom-script',
		get_template_directory_uri() . '/js/custom_script.js',
		array( 'jquery' )
	);
}

add_action( 'admin_menu', 'my_plugin_admin_menu' );
function my_plugin_admin_menu() {
	// Register plugin page
	$page = add_submenu_page(
		'edit.php',                     
		__( 'My plugin', 'myPlugin' ),  
		__( 'My plugin', 'myPlugin' ),  
		'manage_options',               
		'my_plugin-options',            
		'my_plugin_manage_menu'         
	);

	// Connect to a needed hook using the plugin page identifier 
	add_action( 'load-' . $page, 'my_plugin_admin_scripts' );
}

// This function will be called only on the page of the plugin
function my_plugin_admin_scripts() {
	wp_enqueue_script( 'my-plugin-script', plugins_url('/script.js', __FILE__) );
}

function my_plugin_manage_menu() {
	// page code
}

To change file URL when the file content changed in order to disable browser cache, you can specify file version dynamically base on file modification time (filemtime):

add_action('wp_enqueue_scripts', 'my_scripts_method');
function my_scripts_method() {
	wp_enqueue_script( 'custom-script',
		get_template_directory_uri() . '/js/custom_script.js',
		[ 'jquery' ],
		filemtime( get_theme_file_path('js/custom_script.js') )
	);
}

filemtime() performance is very high - on SSD disk 0.5 sec for 50K iterations - it's very fast!

When registering a script, it gets a version (the current version of WordPress, by default): /wp-includes/css/dashicons.min.css?ver=4.9. You can remove it:

// Removing all versions of all scripts (and styles):
add_filter( 'script_loader_src', '_remove_script_version' );
add_filter( 'style_loader_src', '_remove_script_version' );
function _remove_script_version( $src ){
	$parts = explode( '?', $src );
	return $parts[0];
}

Remove only WordPress versions:

add_filter( 'script_loader_src', 'hb_remove_wp_version_from_src' );
add_filter( 'style_loader_src', 'hb_remove_wp_version_from_src' );
function hb_remove_wp_version_from_src( $src ) {
	 global $wp_version;
	 parse_str( parse_url( $src, PHP_URL_QUERY ), $query );
	 if ( ! empty($query['ver']) && $query['ver'] === $wp_version ) {
		  $src = remove_query_arg('ver', $src);
	 }
	 return $src;
}

Read in separate article.