Taxonomy page returns 404 page not found

But when I try

example.com/lesson

I get page not found error.

Like I pointed in my comment to your question, that’s actually how it works, where the term slug needs to be specified in the URL. That’s why example.com/lesson/english (term slug is english) and example.com/lesson/french (term slug is french) work, but not example.com/lesson (term slug not specified).

So one way to make example.com/lesson works, is by setting the has_archive parameter to lesson when registering the CPT memory-cards via register_post_type(). For example:

register_post_type( 'memory-cards', array(
    'labels'       => array(
        'name'          => 'Memory Cards',
        'singular_name' => 'Memory Card',
    ),
    'public'       => true,
    'has_archive'  => 'lesson',
) );

But that also means, example.com/memory-cards, which is the default archive URL for your memory-cards CPT (when has_archive is set to true), would no longer display the CPT archive.

To fix that, or to preserve the default archive URL, you can use add_rewrite_rule() to add the rewrite rules for example.com/lesson, like so:

register_taxonomy( 'lesson', 'memory-cards', array(
    'labels'       => array(
        'name'          => 'Lessons',
        'singular_name' => 'Lesson',
    ),
    'rewrite'      => true,
    // ...other args here...
) );

add_rewrite_rule( 'lesson/?$', 'index.php?post_type=memory-cards', 'top' );
add_rewrite_rule( 'lesson/page/(\d+)/?$', 'index.php?post_type=memory-cards&paged=$matches[1]', 'top' );

The second rewrite rule will handle paged requests such as example.com/lesson/page/2/.