How to save multiple options from a dropdown in user profile

You will need an opening select tag that tells PHP it is an array. Something along the lines of the following should get you on your way…

<?php
// Display Fields 
add_action( 'show_user_profile', 'add_multiple_choice_dropdown ' );
add_action( 'edit_user_profile', 'add_multiple_choice_dropdown ' );

function add_multiple_choice_dropdown ( $user ) {
    $current_selections = get_user_meta( $user->ID, 'multi_dropdown', true );
?>

<h3>Extra profile information</h3>

<table class="form-table"> 
<tr>
<th><label for="multi_dropdown" multiple="multiple">The dropdown with multiple choices</label></th>
<td>
<select name="multi_dropdown[]">
        <option value="first_choice" <?php echo ( !empty( $current_selections ) && in_array( 'first_choice', $current_selections ) ? ' selected="selected"' : '' ) ?>>First Choice</option>
        <option value="second_choice" <?php echo ( !empty( $current_selections ) && in_array( 'second_choice', $current_selections ) ? ' selected="selected"' : '' ) ?>>Second Choice</option>
        <option value="third_choice" <?php echo ( !empty( $current_selections ) && in_array( 'second_choice', $current_selections ) ? ' selected="selected"' : '' ) ?>>Third Choice</option>
</select>
<p class="description">Choose from the options above.</p>
</td>
</tr>
</table>

<?php
}

// Save fields
add_action( 'personal_options_update', 'save_multiple_choices' );
add_action( 'edit_user_profile_update', 'save_multiple_choices' );

function save_multiple_choices( $user_id )    {
    if ( isset( $_POST['multi_dropdown'] ) ) {
        update_user_meta( $user_id, 'multi_dropdown', $_POST['multi_dropdown'] );
    }
}
?>