A wireless range extender is a device that picks up an existing wireless signal on a Wi-Fi network and rebroadcasts it. Range extenders use the same technology as a relay tower in a cellphone network, albeit on a much smaller scale. The devices allow you to spread your Wi-Fi network over a wider area, potentially reaching parts of your house or yard that can't pick up a signal directly from your wireless router.
As long as you are using quality equipment, the main factor affecting a wireless extender's performance is where you place it. Simply put, the stronger the signal the extender receives, the strong the signal it can pass on. This means that you need to find the best balance between siting the extender close enough to the wireless router to pick up a good signal, but close enough to the equipment you want to connect to the network to make using an extender worthwhile.
It takes a split second for the wireless extender to receive, handle, and retransmit data. For purposes such as file downloading or transfers, this isn't a major issue as the overall speed isn't affected significantly. However, it does affect how quickly your computer or other devices are able to respond to a communication from a website: that is, how quickly it can go from a "standing start" to transferring data. This may mean that using a wireless extender can limit your computer's performance in online gaming, where an instant response is important.
All wireless equipment is rated at a particular maximum speed, even if it doesn't always achieve this in practice. The speed is usually expressed as a letter (common variations include B, G, and N, the latter being fastest) but sometimes a number such as 150 or 300 (referring to the maximum speed in megabits per second.) Your wireless network will work only as fast as the slowest component. This means getting the fastest extender available could be a waste, while getting a slow extender could degrade your network performance.
Wi-Fi signals are most commonly carried over the 2.4GHz band. Some Wi-Fi also goes over the 5GHz band: this isn't inherently faster, but because it is less widely used, you are less likely to get interference from your neighbors' equipment. Extenders come in just 2.4GHz or just 5GHz versions, or you can get dual-band equipment that will use whichever band works best. Some dual-band equipment can use both bands simultaneously, potentially doubling data transfer speeds. Make sure your extender and all other Wi-Fi equipment can use the same band; otherwise, you may not be able to establish a connection.