Using widgets from WordPress.com as plugins in WordPress.org

Is that widget available as a plugin for wordpress.org? If so, how do I find it?

Well, it’s not that exactly, but this is about as close as they come: http://wordpress.org/extend/plugins/subscribe2/

if I find the website and am able to download plugin files, where do I save them on my computer? I’ve always just gone through the plugin menus on the wordpress.org dashboard.

You can find Subscribe2 by searching for it at Plugins/Add New in your Dashboard. Alternatively, you can follow this guide to install it manually: http://codex.wordpress.org/Managing_Plugins#Installing_Plugins

I think it adds a widget, which you can add to your sidebar at Appearance/Widgets in your Dashboard, similar to the way WordPress.com does it.

Did you activate the Subscribe2 plugin? It may be a setting that you need to activate within the plugin?