div に隠しオーバーフローを追加すると役立つはずです。このjsFiddle の例を確認してください。基本的な考え方はここで理解できるはずですが、背景とテキストのパディングを微調整する必要があります。
jQuery
$("#div1").hover(function() {
$(this).animate({
height: '300px'
});
}, function() {
$(this).animate({
height: '100px'
});
});
HTML
<div id="div1" style="height:120px; width:206px;overflow:hidden;background: url('http://quiltersbee.com/images/white-postit.png') no-repeat 0 0; ">
<p style="padding-left: 8;padding-right: 8;">This is the paragraph that I would like to be only 190px wide and placed on the background image. It should be left-aligned. </p><br />
<p style="padding-left: 8;padding-right: 8;">This is the paragraph I would like to hide until someone mousesovers the above paragraph. Both paragraphs should be indented so they don't overlap the shading on the background image when this paragraph shows. This paragraph may include a link.</p>
</div>