Hi Gayle,
E-zines and email newsletters are very similar. E-zine is short for email magazine, so an e-zine is typically longer than an email newsletter (or e-newsletter). An e-zine might contain two articles and a few mini sections (like A Resources or Recommendation section, or maybe a Tip of the Month section), while an e-newsletter might just contain one article. If you'd like specific examples of e-zines and e-newsletters, just PM me and I can send some over to you.
As far as setting up an e-newsletter or e-zine, you would first need an autoresponder program (as Diana mentioned) - I personally use iContact and have been very happy with it, but I've also heard good things about Constant Contact too. Then you would choose between either a text e-newsletter or HTML e-newsletter. If you go with a text e-newsletter, you would just type out your content in the autoresponder program and send it off. For HTML newsletters, you can either use the free templates that come with most autoresponder programs or you could have a designer create one specifically for your business (I chose to go that route because it ties in with my branding

). There's really a lot that goes into getting started with an e-newsletter or e-zine...so this is just the tip of the iceberg. There's much more to it than I can get into here, but please feel free to PM me if you'd like me to point you toward some more free resources to help you get started.
Best wishes!