Menu by itself is a taxonomy in WP. It means that you can find all menus in wp_terms table, by running following query:
SELECT *
FROM wp_terms AS t
LEFT JOIN wp_term_taxonomy AS tt ON tt.term_id = t.term_id
WHERE tt.taxonomy = 'nav_menu';
Menu item is custom post type in WP. They are stored in wp_posts table. You can find all of them by using this query:
SELECT *
FROM wp_posts
WHERE post_type="nav_menu_item";
Relations between menus and menu items are stored in wp_term_relationships table. To find all items of specific menu you can use this query:
SELECT p.*
FROM wp_posts AS p
LEFT JOIN wp_term_relationships AS tr ON tr.object_id = p.ID
LEFT JOIN wp_term_taxonomy AS tt ON tt.term_taxonomy_id = tr.term_taxonomy_id
WHERE p.post_type="nav_menu_item"
AND tt.term_id = /*your menu ID*/;
Information about currently select menu is located in wp_options table as serialized PHP array. For example if we use TwentyEleven theme, then we will have a record in wp_options table with option_name column equals to theme_mod_twentyeleven and option_value column equals to ...;s:18:"nav_menu_locations";a:1:{s:7:"primary";i:103;}}. Here you can see that menu with term_id equals to 103 is currently selected as "primary" menu.
The key principle here is that we always have separate options record for each WP theme. Such options have the same name structure: theme_mods_{your-theme-name}.
P.S.: To change current menu in the admin panel, just go to Appearance ยป Menus page and select menu you need in Theme Locations meta box:
