You can configure Contact Enhanced to display a Registration Form, however please keep in mind that this is not the main purpose of Contact Enhanced, therefore the only Form Fields recorded on the User's database table are name, username, -mail and password.
This is something a few users ask for, but we do not encourage/recommend using Contact Enhanced as a registration form. There are many registrations extensions available for Joomla at JED.
Usage
For Joomla 3 and newer:
You'll need to first create a Contact Enhanced Category, Contact and assign form fields to this category. For more information on how to do that please read the Cheat Sheet article first;
Then go to Menus → [a menu type of your preference, Main Menu for instance];
Click on the New button and choose Contact Enhanced → Create Contact Menu item type.
Enable the User Registration option under the "Contact Display Options" tab and hide all information you don't want to display;
Enable (Allow) Create Permission in Contact Enhanced Options for the Public user group;
Below is a video tutorial for Joomla 1.5 and it applies for Contact Enhanced for Joomla 2.5 as well.
Keep in mind this option will not create a Contact in Contact Enhanced. If you want to create a Contact for each user then you might need to read these documentation articles as well: