What is a variant in Page Manager?
I've never been forced to define it before, but here's an attempt at it.

"A variant is a specific response to a request on a Page", in the Page manager. Mostly, using Panels, this involves the rendering of a node or other full page, but not always.

While there are some similarities to "Views displays", unlike Displays, Variants have no inheritance between each other. All variants on any particular Page has the same basic access rules if any, and starts of with whatever context-objects that Page manager derives from the URL, but are not aware of eachother in any way. Based on selection rules, they are evaluated from top to bottom. The first selection rule to return TRUE, gets picked.

A common example is to separate two different content/node-types from each other, allowing you to render them entirely differently. Since each variant may pick its own layout, and place fields/forms/etc in an arbitrary way, the "Article" and "Blog" types may look entirely different. In this case the variants would correspond to node-article.tpl.php and node-blog.tpl.php.

There are various other "Responders" available for Variants that don't involve the use of Panels, such as Contextual administration and plain HTTP Response codes, but I believe this answer is long enough as it is.