The episode you refer to is the the one where Leo and Steve discuss how Opera Mini routes all requests through a proxy which systematically edits the pages you are served, as Steve states:

It is rewriting pages, web pages on the fly, rewriting JavaScript on the fly, essentially turning web pages that were never designed to be seen on a very small screen on a very lightweight and lower powered browser, making them work.

The concern here is that any site that is visited on an Opera Mini browser that the user tries to make secure (by putting https:// at the front of the URL) is made such that it is only secure between Opera’s proxy and the server itself. The information is broadcast in the clear between the proxy and your phone.

I’m not entirely familiar with either Opera Mini or Opera Mobile, but I would recommend doing as I did and reading their respective Wikipedia pages. Of particular interest is the comparison rendering on each page showing how each renders Wikipedia’s main page. With Mobile, it is recognizably the same page, just scaled down to fit on a very narrow screen.

On the other hand, Opera Mini has a much more eye-pleasing view which, I imagine, ignores most of the CSS of the page and distills the site into its core information. Desktop Opera can do something similar, by doing a special fullscreen with shift+F11, which makes the display about 320 pixels wide and does much of this re-processing. From the screenshot, it would appear that Mobile does not do this by default, although there may be a way to configure it to do so.

Incidentally, there’s a bit of an argument back and forth on this. Some people want websites made to display better on handhelds through special stylesheets and such, and some people don’t want special treatment and just want their handheld web browser to be able to handle the full-scale pages. I am told the latter group is fortified in its ranks by Apply iPhone users who have a beefy handheld that is capable of rendering a full-size page, matched with a tweaked build of Safari that makes navigation of such easier.

Moral of the story? Don’t do banking or check email via Opera Mini, because it’ll strip out your SSL connection. For everything else, it’s probably the preferable interface for checking sports scores or using Wikipedia to settle a bet.