add_filter( 'wp_nav_menu_objects', 'wpse16243_wp_nav_menu_objects' );
function wpse16243_wp_nav_menu_objects( $sorted_menu_items )
{
    foreach ( $sorted_menu_items as $menu_item ) {
        if ( $menu_item->current ) {
            $GLOBALS['wpse16243_title'] = $menu_item->title;
            break;
        }
    }
    return $sorted_menu_items;
}

add_filter( 'single_cat_title', 'wpse16243_single_cat_title' );
function wpse16243_single_cat_title( $cat_title )
{
    if ( isset( $GLOBALS['wpse16243_title'] ) ) {
        return $GLOBALS['wpse16243_title'];
    }
    return $cat_title;
}

$cat = get_the_category();
$currentcat = $cat[0];

$currentcat=>cat_ID // Category ID
$currentcat=>cat_name // Category Name (Same as Title)
$currentcat=>category_nicename // Category Slug
$currentcat=>category_description // Category Description
$currentcat=>category_count // Category Count (# of Posts w/ this Category)

add_filter( 'wp_nav_menu_objects', 'wpse16243_wp_nav_menu_objects' );
function wpse16243_wp_nav_menu_objects( $sorted_menu_items )
{
    foreach ( $sorted_menu_items as $menu_item ) {

        if ($menu_item->current ) {
            $GLOBALS['currentMenuTitle'] = $menu_item->title;
            $GLOBALS['currentMenuID'] = $menu_item->ID;



            break;
        }
    }


    return $sorted_menu_items;
}


function get_menu_items_children( $menu_item_id='' )
{
    global $wpdb;

      //     AND meta_key='_menu_item_menu_item_parent'

    $myrows = $wpdb->get_results( "SELECT * FROM $wpdb->postmeta JOIN $wpdb->posts ON $wpdb->postmeta.post_id=$wpdb->posts.id WHERE meta_value = '".$menu_item_id."' AND meta_key='_menu_item_menu_item_parent' ORDER BY $wpdb->posts.menu_order ASC" );


    foreach ( $myrows as $menu_item ) {

        $sc = get_object_vars($menu_item);

        $myrowsb = $wpdb->get_results( "SELECT * FROM $wpdb->postmeta WHERE post_id = '".$sc['post_id']."' AND meta_key='_menu_item_object_id' " );

        foreach ( $myrowsb as $menu_itemb ) {

            $scb = get_object_vars($menu_itemb);
            $pmIDs[] = $scb['meta_value'];

        }

    }


    return $pmIDs;
}

<?php
function my_get_menu_item_name( $loc ) {
    global $post;

    $locs = get_nav_menu_locations();

    $menu = wp_get_nav_menu_object( $locs[$loc] );

    if($menu) {

        $items = wp_get_nav_menu_items($menu->term_id);

        foreach ($items as $k => $v) {
            if ($items[$k]->object_id == $post->ID) {
                $name = $items[$k]->title;
                break;
            }
        }

    }
    return $name;
}

<?php
$menu_name = my_get_menu_item_name( 'NAV_MENU_LOCATION_NAME' ); // eg. 'primary'

echo $menu_name;