Creating a custom settings page includes the combination of: creating an administration menu, using Settings API and Options API.

The example below can be used for quick reference on these topics by following the comments.

Complete Example

Complete example which adds a Top-Level Menu named WPOrg, registers a custom option named wporg_options and performs the CRUD (create, read, update, delete) logic using Settings API and Options API (including showing error/update messages).

/**
 * @internal never define functions inside callbacks.
 * these functions could be run multiple times; this would result in a fatal error.
 */

/**
 * custom option and settings
 */
function wporg_settings_init() {
	// Register a new setting for "wporg" page.
	register_setting( 'wporg', 'wporg_options' );

	// Register a new section in the "wporg" page.
	add_settings_section(
		'wporg_section_developers',
		__( 'The Matrix has you.', 'wporg' ), 'wporg_section_developers_callback',
		'wporg'
	);

	// Register a new field in the "wporg_section_developers" section, inside the "wporg" page.
	add_settings_field(
		'wporg_field_pill', // As of WP 4.6 this value is used only internally.
		                        // Use $args' label_for to populate the id inside the callback.
			__( 'Pill', 'wporg' ),
		'wporg_field_pill_cb',
		'wporg',
		'wporg_section_developers',
		array(
			'label_for'         => 'wporg_field_pill',
			'class'             => 'wporg_row',
			'wporg_custom_data' => 'custom',
		)
	);
}

/**
 * Register our wporg_settings_init to the admin_init action hook.
 */
add_action( 'admin_init', 'wporg_settings_init' );


/**
 * Custom option and settings:
 *  - callback functions
 */


/**
 * Developers section callback function.
 *
 * @param array $args  The settings array, defining title, id, callback.
 */
function wporg_section_developers_callback( $args ) {
	?>
	<p id="<?php echo esc_attr( $args['id'] ); ?>"><?php esc_html_e( 'Follow the white rabbit.', 'wporg' ); ?></p>
	<?php
}

/**
 * Pill field callbakc function.
 *
 * WordPress has magic interaction with the following keys: label_for, class.
 * - the "label_for" key value is used for the "for" attribute of the <label>.
 * - the "class" key value is used for the "class" attribute of the <tr> containing the field.
 * Note: you can add custom key value pairs to be used inside your callbacks.
 *
 * @param array $args
 */
function wporg_field_pill_cb( $args ) {
	// Get the value of the setting we've registered with register_setting()
	$options = get_option( 'wporg_options' );
	?>
	<select
			id="<?php echo esc_attr( $args['label_for'] ); ?>"
			data-custom="<?php echo esc_attr( $args['wporg_custom_data'] ); ?>"
			name="wporg_options[<?php echo esc_attr( $args['label_for'] ); ?>]">
		<option value="red" <?php echo isset( $options[ $args['label_for'] ] ) ? ( selected( $options[ $args['label_for'] ], 'red', false ) ) : ( '' ); ?>>
			<?php esc_html_e( 'red pill', 'wporg' ); ?>
		</option>
 		<option value="blue" <?php echo isset( $options[ $args['label_for'] ] ) ? ( selected( $options[ $args['label_for'] ], 'blue', false ) ) : ( '' ); ?>>
			<?php esc_html_e( 'blue pill', 'wporg' ); ?>
		</option>
	</select>
	<p class="description">
		<?php esc_html_e( 'You take the blue pill and the story ends. You wake in your bed and you believe whatever you want to believe.', 'wporg' ); ?>
	</p>
	<p class="description">
		<?php esc_html_e( 'You take the red pill and you stay in Wonderland and I show you how deep the rabbit-hole goes.', 'wporg' ); ?>
	</p>
	<?php
}

/**
 * Add the top level menu page.
 */
function wporg_options_page() {
	add_menu_page(
		'WPOrg',
		'WPOrg Options',
		'manage_options',
		'wporg',
		'wporg_options_page_html'
	);
}


/**
 * Register our wporg_options_page to the admin_menu action hook.
 */
add_action( 'admin_menu', 'wporg_options_page' );


/**
 * Top level menu callback function
 */
function wporg_options_page_html() {
	// check user capabilities
	if ( ! current_user_can( 'manage_options' ) ) {
		return;
	}

	// add error/update messages

	// check if the user have submitted the settings
	// WordPress will add the "settings-updated" $_GET parameter to the url
	if ( isset( $_GET['settings-updated'] ) ) {
		// add settings saved message with the class of "updated"
		add_settings_error( 'wporg_messages', 'wporg_message', __( 'Settings Saved', 'wporg' ), 'updated' );
	}

	// show error/update messages
	settings_errors( 'wporg_messages' );
	?>
	<div class="wrap">
		<h1><?php echo esc_html( get_admin_page_title() ); ?></h1>
		<form action="options.php" method="post">
			<?php
			// output security fields for the registered setting "wporg"
			settings_fields( 'wporg' );
			// output setting sections and their fields
			// (sections are registered for "wporg", each field is registered to a specific section)
			do_settings_sections( 'wporg' );
			// output save settings button
			submit_button( 'Save Settings' );
			?>
		</form>
	</div>
	<?php
}