Registering customers on wordpress sites

There are quite good number of Plugins. Go to plugins page to search as per your need.

http://wordpress.org/extend/plugins/

http://wordpress.org/extend/plugins/free-wp-membership/ is possibly an example of what you need.

I do not know about any rock solid free option, but there are paid plugins like WPMU Membership Plugin :

http://premium.wpmudev.org/project/membership/

Or you can use Gravity Form with various addons :

http://www.gravityforms.com/