Example on how to download, save and insert image as a attachment to Wordpress from a web url.

save_image.php

<?php

/**
* Create image attachment using a web url.
*
* @param String url Url path to the image we want to add (http://test.com/img.jpg)
* @param String filename Name of file ex: image.jpg. Will otherwise be loaded by header.
* @param int post_id The post where image should be associated if featured
*        is true. (Optiona)
* @param Boolean featured Whether a file should be featured or not. (Optional)
* @return int Id for the created attachment.
*/
function save_image($url, $filename = NULL, $post_id = 0, $featured = false) {
    // Read file
    if ($filename == NULL) {
        // Loop through headers and and find filename.
        $headers = get_headers($url);
        foreach ($headers as $header_key => $header_data) {
            $matches = array();
            $result = preg_match("/filename=\"(.*)\"$/i", $header_data,
                $matches);

            if (count($matches[1])) {
                $filename = $matches[1];
            }
        }

        // If no filename is found, use basename
        if ($filename == NULL) {
            $filename = basename($url);
        }
    }
    
    $filename = sanitize_file_name($filename);

    // Load file content
    $data = file_get_contents($url);
    if ($data == false) {
        return 0;
    }

    // Save file to disk
    $upload_dir = array(
        "path" => "./",
        "url" => ""
    );

    $save_path = $upload_dir["path"]."/".$filename;
    $save_url = $upload_dir["url"].$filename;

    $response = file_put_contents($save_path, $data);

      // Add attachment
    $filetype = wp_check_filetype($save_path, null );

    $attachment = array(
        "guid" => $save_url,
        "post_mime_type" => $filetype["type"],
        "post_title" => $filename,
        "post_content" => "",
        "post_status" => "inherit"
    );

    $attachment_id = wp_insert_attachment($attachment, $save_path, 0);

    // Generate metadata
    require_once(ABSPATH . 'wp-admin/includes/image.php');
    $attachment_data = wp_generate_attachment_metadata($attachment_id,
        $save_path);
    $metadata = wp_update_attachment_metadata($attachment_id,
        $attachment_data);

    // Add as a featured image
    if ($featured && $metadata) {
        update_post_meta($post_id, '_thumbnail_id', $attachment_id);
    }

    return $attachment_id;
}

save_image("http://yourfileurl.com/image.jpg");
?>