Pekka has explained it in theory in his answer. Based on his answer, and my answer to another question about the > combinator, I'll provide an illustration, modified to address this question.
Consider the following block of HTML, and your example CSS selectors. I use a more elaborate example so I can show you the difference between both of your selectors:
<div>
<p>The first paragraph.</p> <!-- [1] -->
<blockquote>
<p>A quotation.</p> <!-- [2] -->
</blockquote>
<div>
<p>A paragraph after the quotation.</p> <!-- [3] -->
</div>
</div>
Which <p>s are selected by which selectors?
First off, all of them match div p because they are <p> elements situated anywhere within a <div> element.
That makes div > p more specific, which begs the next question:
Which <p>s are selected by div > p?
Selected
This paragraph <p> is a child, or a direct descendant, of the outermost <div>. That means it's not immediately contained by any other element than a <div>. The hierarchy is as simple as the selector describes, and as such it's selected by div > p.
Not selected
This <p> is found in a <blockquote> element, and the <blockquote> element is found in the outermost <div>. The hierarchy would thus look like this:
div > blockquote > p
As the paragraph is directly contained by a blockquote, it's not selected by div > p. However, it can match blockquote > p (in other words, it's a child of the <blockquote>).
Selected
This paragraph lives in the inner <div>, which is contained by the outer <div>. The hierarchy would look like this:
div > div > p
It doesn't matter if there are more <div>s nested within each other, or even if the <div>s are contained by other elements. As long as this paragraph is directly contained by its own <div>, it will be selected by div > p.