How does the Drupal session system work?
Drupal has a pluggable session handler, however it generally uses a cookie + database session model by default.

The sessions table in the database stores the session details for the users. The session id is stored and matched from a browser cookie.

The settings for session duration can be found in settings.php, in particular:

/**
 * Set session lifetime (in seconds), i.e. the time from the user's last visit
 * to the active session may be deleted by the session garbage collector. When
 * a session is deleted, authenticated users are logged out, and the contents
 * of the user's $_SESSION variable is discarded.
 */
ini_set('session.gc_maxlifetime', 200000);

/**
 * Set session cookie lifetime (in seconds), i.e. the time from the session is
 * created to the cookie expires, i.e. when the browser is expected to discard
 * the cookie. The value 0 means "until the browser is closed".
 */
ini_set('session.cookie_lifetime', 2000000);


You can adjust those settings to affect the session lifetime. Indeed, the "remember me" module uses the session.cookie_lifetime setting to apply its functionality.