To add a custom field, from your site’s admin panel you can setup a user field: customize > user fields
This will create a field for a user to fill out in their preferences, and it can optionally be shown on the user card.
If you’re looking to hide information on the user card, the easiest way is by adding custom CSS to your site’s theme.
For more advanced customization you could override the user-card contents template, which looks like this by default. Overriding handlebars templates is covered in the advanced section of Developer’s guide to Discourse Themes