Like he said - make sure you purchase from a reputable company. For me, that would mean I only purchase things online from Amazon, Ebay, direct from the manufacturer, or from the website of a local retailer from whom I can go pick up the paid for item in person.
Also, a refurbished phone is a phone that has been built with used parts. To elaborate; if you paid for an insurance plan when you bought the phone, you would take that phone to the store you bought it from and they would give you a new one.
When the broken phone gets to the manufacturer, they take it apart and put any parts that aren't broken aside. When they have enough parts to build a new phone of the same model (or another model utilizing some of the same hardware), they do - and it is sold as a refurbished phone.
I once purchased a refurbished laptop and had no issues with it for about two years. Shortly after that, the fan died and the CPU withered and burned. Your experience may vary.