What are refurbished and open box items?
Refurbished products are usually customer returns due to a defect of some sort. These products are usually fixed and resold as refurbished at a lower cost. Refurbished items are usually sold online.
An open box item is one that has had the manufacturer's packaging opened for whatever reason. These products could be display models, customer returns or just items that had packaging damaged in transit and the store couldn't sell it as 'new in the box'. Open box items are usually sold in a store.
My personal experience with refurbished and open box items
Personally, I’ve bought two open-box HDTVs and one refurbished PC. Knock on wood but I’ve never had a problem with either HDTV or the PC. I chose not to buy the extended warranty on the two HDTVs or the PC because I was satisfied with the manufacturer's warranty.
However, I must warn that I scrutinized the decision and recommend you do the same for your situation.
Why buy refurbished or open box items?
One school of thought is that the refurbished product already had the defect identified and fixed. Well, hopefully fixed, which is why you should always buy refurbished through a reputable dealer with a warranty attached to the product.
When buying refurbished, I prefer to buy local but if I were to buy online then I would only go to a company that had an explicit warranty statement and return policy.
As far as when going into a store - ask your local electronics salesperson if their store sells refurbished or open-box items.
Other information to consider
While there can be huge savings when buying refurbished or open box items online, keep in mind that shipping charges could offset the cost savings of buying from your local store.
About.com Guide to Home Theater, Robert Silva, wrote a pretty extensive article about buying refurbished items (read article), which I recommend reading to gain further insight.