If I Google "non-strict html" I get surprisingly few hits. If I Google the same and limit the search to htmlhelp.com I don't get any hits that explain what it is--this site does list certain tags and attributes as "non-strict" but I can't find an explanation of what that means. My feeling is that it is something that is somehow only compatible in quirks mode, and shouldn't be used in standards mode. But I have a document that uses some of this non-strict HTML in a strict document, and I'm yet to be forced to regret it. So what is non-strict HTML then?

Standards/Strict mode doesn't require a a Strict doctype. The doctype switching is a whole other ball game.

"Non-strict" probably refers to that the element or attribute in question isn't allowed in (X)HTML Strict. It could be allowed in (X)HTML Transitional or it's totally non standard.

You could say that Strict is core HTML. Transitional allows deprecated, mostly presentational, elements and attributes that Strict doesn't. So for example FONT could be said to be non-strict.

See also http://htmlhelp.com/tools/validator/doctype.html which says

So all this still begs the question: why have I been able to use non-strict elements and attributes in a strict document? What should the consequences of such a thing be?

Because browsers don't care. They don't use the doctype at all other than for this doctype switching crazy (the rendering modes). Had you tried to validate your HTML you would have been told about the non-strict stuff.