1) Is the item in stock?
While this may seem very simple, it is quite often the case that the reason products aren't displaying as as simple as the product being out of stock.
If the product is a complex product type, you will also need to check the stock status of the children products.
2) What is the Visibility & Status of the product?
Like stock, the visibility and status of the product is another good initial check. Is the product enabled, and set to a visibility that would return results from a search or catalog navigation?
3) Is the product assigned to the Website / Category where you are expecting to see it?
If you have multiple websites (and even if you don't), is the product assigned to the website and categories where you are expecting to see it?
4) Could the problem simply be an issue with search? Can you reach the product via the products direct URL? By searching for the SKU or product Name?
Looking at the products record in the backend, if you copy and paste the products URL, can you reach the product page this way? If you search for the products SKU or Product Name is the product returned as a result? Checks like these can help understand whether the issue might be one of search.
5) Are cache and/or indexes out of date?
If you navigate to System > Cache Management, are there cache types out of date? If you navigate to System > Index Management, likewise are there any indexes out of date? An important consideration with Indexes is that Enterprise Edition schedules indexes in, so they shouldn't have to be manually re-indexed, but will rather run at the scheduled time.
To learn more about Indexes and Cache, please see the below links:
If you have tried all of the above and are still not seeing products where you are expecting, please do raise a ticket with us as there might be further or a more complex issue here worth investigating further.