Adds pagination to WordPress without a plugin (taken from HTML5-Blank-WordPress-Theme)

wp-pagination.php

// Add to functions.php
// Pagination for paged posts, Page 1, Page 2, Page 3, with Next and Previous Links, No plugin
function html5wp_pagination()
{
    global $wp_query;
    $big = 999999999;
    echo paginate_links(array(
        'base' => str_replace($big, '%#%', get_pagenum_link($big)),
        'format' => '?paged=%#%',
        'current' => max(1, get_query_var('paged')),
        'total' => $wp_query->max_num_pages
    ));
}

add_action('init', 'html5wp_pagination'); // Add our HTML5 Pagination

// Add to index.php
<?php html5wp_pagination(); ?>

If you add some explanation about customizing the format that would be helpful.