Creating a Front-end based User Search

As you haven’t offered the details on what you’re after, I’ll try to grab them all very briefly.

Use the API – public WP_User_Query API functions

Basically get_user_by() should be enough for you. Let’s say you fire of your form and the input field name was user_id. So you’d just retrieve the value from the $_REQUEST (or $_GET) and then do a simple query.

$user = get_user_by( 'user_id', $_REQUEST['user_id'] );

Advanced Task: Meta User Query

If you’re going to do more complex requests (for maybe only a partly request that covers more than one user as possible result), then it’s time to call get_users(), which is the default API wrapper for new WP_User_Query.

A meta query could look like this:

$args = array(
    'meta_key' => 'custom-usermeta',
    'meta_value' => $cat_id,
    'meta_compare' => '='
);
$users = get_users( $args );

AJAX Search

If you need AJAX search functionality, just take a look at the answers filed under ajax to get around it – no reason repeating it here.

I have asked a question about autocomplete some time ago and covers the most important parts.

// jQuery autocomplete - already bundled with WP core
function wpse71343_scripts()
{
    wp_enqueue_script( 'jquery-ui-autocomplete' );
}
add_action( 'wp_enqueue_scripts', 'wpse71343_scripts' );