The other answers on this page are out of date, but the question is still valid.
q element is an inline element and should be used like so (ie. no block elements inside it):
In the words of <cite>Charles Bukowski</cite> -
<q>An intellectual says a simple thing in a hard way.
An artist says a hard thing in a simple way.</q>
<q>This is correct, said Hillary.</q> is a quote from the
popular daytime TV drama <cite>When Ian became Hillary</cite>.
q element should not be placed inside a
blockquote element, as it would be redundant -- both denote a quote.
blockquote is a block element, allowing other block elements to be placed inside:
<p>My favorite book is <cite>At Swim-Two-Birds</cite>.</p>
- <cite>Mike Smith</cite>
<cite> is slightly more complicated. It's an inline element but it depends which HTML spec you're following. The W3C states that it may contain a URL, a title of a work (eg. book title, film title, etc.), or an author's name.
The WHATWG states that it may only contain a URL or a title of a work, and so not a person's name.
This is a valid WHATWG usage:
<p>The truth may be puzzling. It may take some work to grapple with.
It may be counterintuitive. It may contradict deeply held
prejudices. It may not be consonant with what we desperately want to
be true. But our preferences do not determine what's true.</p>
<figcaption>Carl Sagan, in "<cite>Wonder and Skepticism</cite>", from
the <cite>Skeptical Inquirer</cite> Volume 19, Issue 1 (January-February