// This goes in function.php

class My_Walker extends Walker_Nav_Menu
{
	function start_el(&$output, $item, $depth, $args) {
		global $wp_query;
		$indent = ( $depth ) ? str_repeat( "\t", $depth ) : '';

		$class_names = $value = '';

		$classes = empty( $item->classes ) ? array() : (array) $item->classes;

		$class_names = join( ' ', apply_filters( 'nav_menu_css_class', array_filter( $classes ), $item ) );
		$class_names = '';

		$output .= $indent . '<li id="menu-item-'. $item->ID . '"' . $value . $class_names .'>';

		$attributes  = ! empty( $item->attr_title ) ? ' title="'  . esc_attr( $item->attr_title ) .'"' : '';
		$attributes .= ! empty( $item->target )     ? ' target="' . esc_attr( $item->target     ) .'"' : '';
		$attributes .= ! empty( $item->xfn )        ? ' rel="'    . esc_attr( $item->xfn        ) .'"' : '';
		$attributes .= ! empty( $item->url )        ? ' href="'   . esc_attr( $item->url        ) .'"' : '';

		$item_output = $args->before;
		$item_output .= '<a'. $attributes .'>';
		$item_output .= $args->link_before . apply_filters( 'the_title', $item->title, $item->ID ) . $args->link_after;
		$item_output .= '<span>' . $item->description . '</span>';
		$item_output .= '</a>';
		$item_output .= $args->after;

		$output .= apply_filters( 'walker_nav_menu_start_el', $item_output, $item, $depth, $args );
	}
}

// Then Display the Menu anywhere

	<?php
	$walker = new My_Walker; 
	wp_nav_menu( array( 
		'container_id' => 'mainmenu', 
		'menu_class' => 'sf-menu',
		'walker' => $walker
		)); 
	?>

class My_Walker extends Walker_Nav_Menu
{
	function start_el(&$output, $item, $depth, $args) {
		global $wp_query;
		$indent = ( $depth ) ? str_repeat( "\t", $depth ) : '';

		$class_names = $value = '';

		$classes = empty( $item->classes ) ? array() : (array) $item->classes;

		$class_names = join( ' ', apply_filters( 'nav_menu_css_class', array_filter( $classes ), $item ) );
		$class_names = '';

		$output .= $indent . '<li id="menu-item-'. $item->ID . '"' . $value . $class_names .'>';

		$attributes  = ! empty( $item->attr_title ) ? ' title="'  . esc_attr( $item->attr_title ) .'"' : '';
		$attributes .= ! empty( $item->target )     ? ' target="' . esc_attr( $item->target     ) .'"' : '';
		$attributes .= ! empty( $item->xfn )        ? ' rel="'    . esc_attr( $item->xfn        ) .'"' : '';
		$attributes .= ! empty( $item->url )        ? ' href="'   . esc_attr( $item->url        ) .'"' : '';

		$item_output = $args->before;
		$item_output .= '<a'. $attributes .'>';
		$item_output .= $args->link_before . apply_filters( 'the_title', $item->title, $item->ID ) . $args->link_after;
		$item_output .= '<span>' . $item->description . '</span>';
		$item_output .= '</a>';
		$item_output .= $args->after;

		$output .= apply_filters( 'walker_nav_menu_start_el', $item_output, $item, $depth, $args );
	}
}

// Thene Display the Menu

	<?php
	$walker = new My_Walker; 
	wp_nav_menu( array( 
		'container_id' => 'mainmenu', 
		'menu_class' => 'sf-menu',
		'walker' => $walker
		)); 
	?>

Add Description To the Menu Using wp_nav_menu

PHP