Its pretty simple - you have to do edit your Home page. Go to your Magento admin, login, then go CMS - Manage Pages. Edit your Home page, and in the place you wanna have product list displayed put this row:

{{block type="catalog/product_list"
category_id="1"
template="catalog/product/list.phtml"}}

Where you have to change the category_id attribute to category id you want to display. You can find this id in heading, if you editing any category.