<p> is the tag to open a paragraph.

</p> is the tag to close a  paragraph.

<p /> is the tag to both open and close a paragraph with no contents in the paragraph.  It basically results in a blank line.

Almost all HTML tags can be opened and closed in one tag if they’re contents are empty.  Another example might be:

<td /> - A table data cell with no contents

<tr /> - A table row with no data cells

Some tags should technically ALWAYS be written this way because they can’t contain data:

<br /> - A line break

<hr /> - A horizontal rule (line)

This is technically a specification of XHTML, which is (more or less) just properly formed HTML.  Web browsers will interpret a <p> without a closing </p> and they’ll interpret a <br> the same as a <br />.  It’s just a specification of XHTML that every opening tag should have a closing tag (or open and close itself, as in <br />).

More information is available on XHTML and HTML at http://www.w3.org/MarkUp/2004/xhtml-faq