In order to show description show in the front page under the category do the following.
In your main.tpl (in your template dir) find the following code (or close to the follwoing)
<div class="cat">
<a href="{if $smarty.const.ENABLE_REWRITE}{$cat.TITLE_URL|escap e}/{else}index.php?c={$cat.ID}{/if}">{$cat.TITLE|escape}</a>{if $smarty.const.CATS_COUNT}<span>({$cat.COUNT})</span>{/if}
<div class="desc" style="width:275px">{$cat.DESCRIPTION}</div>
</div>
Add the underlined code above. You can remove the class and style portion and style it as you see fit.