How do I use wp_register_script() and wp_enqueue_script() functions

I have a plugin that uses jQuery, when activated, it breaks my template.
The plugin does not work unless I comment out the following in the footer.php
<script src="http://ajax.googleapis.com/ajax/libs/jquery/1.11.1/jquery.min.js"></script>

When I comment this jquery out the template breaks. When it’s not commented out, the template is fine, but the plugin does not work.

This is in my functions.php

add_filter( 'wp_enqueue_scripts', 'change_default_jquery', PHP_INT_MAX );

function change_default_jquery( ){
    //wp_dequeue_script( 'jquery');
    //wp_deregister_script( 'jquery');
}

The main problem is that there are two jQuery files with the same version loaded. I can’t find the call to jquery.js?ver1.11.1 that shows up in the header. I want to add script, and need to use wp_register_script() and wp_enqueue_script() functions.

I found a solution, but i don’t understand how it works.

function wptuts_scripts_with_jquery()
{
    // create array of all scripts
    $scripts=array('custom-script'=>'/js/custom-script.js',
                  'custom-script1'=>'/js/custom-script1.js',
                  'custom-script2'=>'/js/custom-script2.js');
    foreach($scripts as $key=>$sc)
    {
       wp_register_script( $key, get_template_directory_uri() . $sc, array('jquery') );
       wp_enqueue_script( $key );
    }
}
add_action( 'wp_enqueue_scripts', 'wptuts_scripts_with_jquery' );

How do I solve this problem?