<p>I use div tags to define areas within my web pages.  I set all the obvious things like background, size, padding, etc.  But it is all very square.</p>

<p>How can I use <strong>only</strong> CSS to round the corners?</p>

You would use the border-radius property.  However, this is only supported in CSS3, which no browser implements yet.  If you only need it to work in a couple browsers you could use -webkit-border-radius and -moz-border-radius which would let it work in Safari and Firefox respectively.
If you are not opposed to using images.  Here's s method I came up with for
making rounded borders
.