Use a widget, shortcode, and/or template tags to easily retrieve and display custom field values for the currently logged in (or any specific) user.
The user profile page within WordPress provides inputs for a handful of user custom fields (first_name, last_name, aim, yim, jabber, description, etc). However, you're probably more interested in creating your own user custom fields. In that case, you'll have to use another plugin to store custom fields for users, or directly use WordPress functions manually.
When on the Write or Edit admin pages for a page or post, find the "Screen Options" link near the upper right-hand corner. Clicking it slides down a panel of options. In the "Show on screen" section, uncheck the checkbox labeled "Get User Custom Field Values - Shortcode". This must be done separately for posts and for pages if you want the shortcode builder disabled for both sections.