There are plenty of Newsletter modules / plugins for WordPress. If you are using a service like CampaignMonitor, Aweber or MailChimp there are great plugins for integrate your WordPress page with these. But here I will list the best stand alone plugins for addidng a newsletter/mailinglist to WordPress:

ALO EasyMail Newsletter:

http://wordpress.org/extend/plugins/alo-easymail/screenshots/
Looks great and handles registred members as well as not-registered users. You can use multiple lists (mailingslists) and integrate usernames etc. into the actual email. Simple statistics shows how many have opened your email etc.. The newsletter software can be easily translated to your preferred language.

MeeNews Newsletter Plugin

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

This one is a bit more advanced and lets you handle templates / layout etc. for your newsletter in a more advanced way. Seems to be free for non-commercial use only though.

Newsletter

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

Very easy to use! Does not handle multiple lists though. To get advanced features tou need to upgrade to Newsletter Pro (paid of course).. It is easily inserted anywhere simply by writing  [ newsletter ] (without inner spaces and lower case). It’s easy to change between single or double opt-in – and you can handle the responses send when new peebs subscribe.

WP Responder Email Newsletter and Autoresponder Plugin

http://wordpress.org/extend/plugins/wp-responder-email-autoresponder-and-newsletter-plugin/

If you actually need a Auto-responder system instead of a normal newsletter system, this is the way to look!
This basically allows you to schedule emails to be send after a specific number of days after the subscriber signs up.

Update: This also looks good: http://www.mailpress.org/

AND here you get a comparison: http://grahamstoney.com/blogging/comparison-wordpress-email-newsletter-subscription-plugins