There can be various reasons why your Magento categories and products are not showing. Let's start walking through all the options that make your categories and products appear on your Magento site.
Magento catalog categories (or product categories) can be managed through the page Manage categories in the Magento Admin Panel. For each category you can determine the option Is Active. But an active category does not neccessarily show up in the shop. It has to be part of a Root Catalog category that is actually assigned to a specific Magento scope.
One other tip is that a Magento category appears (in most circumstances) only if there are products assigned to it. An empty category will for instance not be shown in the default category-menu.
First of all, each category is ment to be either a Root Category (also called Root Catalog) or a subcategory of such a Root Category. See a Root Category as the starting point of your webshop. This also means that your webshop needs to be configured with this Root Category. If you are just working with the original Root Catalog that Magento created for you, this is not something you need to worry about - just make sure your subcategories are positioned as children of this Root Catalog. But if you want to create your own Root Categories, make sure these Root Categories are actually used by a Store (System > Manage Stores).
To make products appear on the site, the first step is to make sure such a product is available through a category. But there are more things to think about. The product status can be set to "Enabled" or "Disabled". The product can also be made visible through the "Catalog", or only through "Search", or both - but it can also be made invisible, which means that though the product is enabled it's not appearing anywhere on the site.
Because a single product could appear not only in the catalog (meaning: product categories) but also through search, you also need to determine if a product is part of a specific store anyway. This is done through the tab "Websites".
To give you a quick overview, a products appearance in the shop depends on the following settings: