Text emails are the simple emails that contains text as its content. You will see the plain text with hightlighted links. That is it! No colors or no images in text emails.
See the following image of text email format:
It is a promotional email sent to a list of an ezine publisher. Simple text is displaying.

Html emails are the emails with html in them. They look like web pages with colors, images and pretty font etc.
See the following image of html email format:
It is the html version of the above text email. See the contrast between the two email formats. Html emails are eye catching.

Most important things you should note before sending the html emails:
* Whether your subscriber likes it or not - some people don't like to receive html emails because of the bias that these emails contain viruses, attachments, background scripts running etc.
* Limitations of your subscriber email client - not all email clients display html.
Or your subscribers may intentionally turn off html display. In these cases the html emails won't display in a proper way.
What is the best thing to do?
* Send text emails as default. You will not have any problems with it.
* Give option to the subscriber. On your subscription form use a format option to choose from, whether text or html format. The problem with this is you need to create two emails one for subscribers who opted for text and one for subscribers who opted for html.
Email marketers differ in their opinion on email formats. It would be their personal preference most of the times. Personally I'd like to see text emails in my inbox, so I prefer sending text emails to clients and subscribers.