Wordpress - What is the preferred way to add custom javascript files to the site?

Use wp_enqueue_script() in your theme

The basic answer is to use wp_enqueue_script() in an wp_enqueue_scripts hook for front end admin_enqueue_scripts for admin. It might look something like this (which assumes you are calling from your theme's functions.php file; note how I reference the stylesheet directory):

<?php
add_action( 'wp_enqueue_scripts', 'mysite_enqueue' );

function mysite_enqueue() {
  $ss_url = get_stylesheet_directory_uri();
  wp_enqueue_script( 'mysite-scripts', "{$ss_url}/js/mysite-scripts.js" );
}

That's the basics.

Pre-defined and Multiple Dependent Scripts

But let's say you want to include both jQuery and jQuery UI Sortable from the list of default scripts included with WordPress and you want your script to depend on them? Easy, just include the first two scripts using the pre-defined handles defined in WordPress and for your script provide a 3rd parameter to wp_enqueue_script() which is an array of the script handles used by each script, like so:

<?php
add_action( 'wp_enqueue_scripts', 'mysite_enqueue' );

function mysite_enqueue() {
  $ss_url = get_stylesheet_directory_uri();
  wp_enqueue_script( 'mysite-scripts', "{$ss_url}/js/mysite-scripts.js", array( 'jquery', 'jquery-ui-sortable' ) );
}

Scripts in a Plugin

What if you want to do it in a plugin instead? Use the plugins_url() function to specify the URL of your Javascript file:

<?php
define( 'MY_PLUGIN_VERSION', '2.0.1' );
add_action( 'wp_enqueue_scripts', 'my_plugin_enqueue' );

function my_plugin_enqueue() {
  wp_enqueue_script( 'my-script', plugins_url('/js/my-script.js',__FILE__), array('jquery','jquery-ui-sortable'), MY_PLUGIN_VERSION );
}

Versioning your Scripts

Also note that above we gave our plugin a version number and passed it as a 4th parameter to wp_enqueue_script(). Version number is output in source as query argument in URL to script and serves to force browser to re-download possibly cached file if version changed.

Load Scripts only on pages where needed

The 1st rule of Web Performance says to Minimize HTTP Requests so whenever possible you should limit the scripts to load only where needed. For example if you only need your script in the admin limit it to admin pages using the admin_enqueue_scripts hook instead:

<?php
define( 'MY_PLUGIN_VERSION', '2.0.1' );
add_action( 'admin_enqueue_scripts', 'my_plugin_admin_enqueue' );

function my_plugin_admin_enqueue() {
  wp_enqueue_script( 'my-script', plugins_url( '/js/my-script.js', __FILE__ ), array( 'jquery', 'jquery-ui-sortable' ), MY_PLUGIN_VERSION );
}

Load Your Scripts in the Footer

If your scripts is one of those that need to be loaded into the footer there is a 5th parameter of wp_enqueue_script() that tells WordPress to delay it and place it in the footer (assuming your theme did not misbehaved and that it indeed calls the wp_footer hook like all good WordPress themes should):

<?php
define( 'MY_PLUGIN_VERSION', '2.0.1' );
add_action( 'admin_enqueue_scripts', 'my_plugin_admin_enqueue' );

function my_plugin_admin_enqueue() {
  wp_enqueue_script( 'my-script', plugins_url( '/js/my-script.js', __FILE__ ), array( 'jquery', 'jquery-ui-sortable' ), MY_PLUGIN_VERSION, true );
}

Finer Grained Control

If you need finer grained control than that Ozh has a great article entitled How To: Load Javascript With Your WordPress Plugin that details more.

Disabling Scripts to Gain Control

Justin Tadlock has a nice article entitled How to disable scripts and styles in case you want to:

1. Combine multiple files into single files (mileage may vary with JavaScript here).
2. Load the files only on the pages we’re using the script or style.
3. Stop having to use !important in our style.css file to make simple CSS adjustments.

Passing Values from PHP to JS with wp_localize_script()

On his blog Vladimir Prelovac has a great article entitled Best practice for adding JavaScript code to WordPress plugins where he discusses using wp_localize_script() to allow you to set the value of variables in your server-side PHP to be later used in your client-side Javascript.

Really Fine Grained Control with wp_print_scripts()

And finally if you need really fine-grained control you can look into wp_print_scripts() as discussed on Beer Planet in a post entitled How To Include CSS and JavaScript Conditionally And Only When Needed By The Posts.

Epiloque

That's about it for Best Practices of including Javascript files with WordPress. If I missed something (which I probably have) be sure to let me know in the comments so I can add an update for future travelers.

To compliment Mikes wonderful illustration of using enqueues i just wish to point out that scripts included as dependancies do not need to be enqueued...

I'll use the example under the Scripts in a Plugin heading in Mike's answer.

define('MY_PLUGIN_VERSION','2.0.1');

add_action( 'wp_enqueue_scripts', 'my_plugin_enqueue_scripts' );
function my_plugin_enqueue_scripts() {
    wp_enqueue_script('jquery');
    wp_enqueue_script('jquery-ui-sortable');
    wp_enqueue_script('my-script',plugins_url('/js/my-script.js',__FILE__),array('jquery','jquery-ui-sortable'),MY_PLUGIN_VERSION);
}

This can be trimmed down to read..

define('MY_PLUGIN_VERSION','2.0.1');

add_action( 'wp_enqueue_scripts', 'my_plugin_enqueue_scripts' );
function my_plugin_enqueue_scripts() {
    wp_enqueue_script('my-script', plugins_url( '/js/my-script.js', __FILE__ ), array('jquery','jquery-ui-sortable'), MY_PLUGIN_VERSION);
}

When you set dependancies, WordPress will find and enqueue them ready for your script, so you don't need to make any independent calls for these scripts.

Additionally, some of you may be wondering if setting a multiple dependancies might cause scripts to output in the wrong order, here let me give an example

wp_enqueue_script( 'first-script', 'example/path/example_script_1.js', array('second-script','third-script') );

If script two were dependant on script three(ie. script three should load first), this wouldn't matter, WordPress will determine the enqueue priorities for those scripts and output them in the correct order, in short, it all gets worked out automagically for you by WordPress.