Here is a very simplified attempt to answer.
Extenders are really just another name for repeaters. They sit at a distance from the wireless radio of say a router and receive the signal then retransmit it on another channel to increase the signal strength at more distant areas. They do halve your bandwidth, and I generally do not recommend their use.
APs are the most useful way to extend a wireless signal and you can use more than one (many in fact if properly laid out). They must be connected to the router LAN port to LAN port (but can have intervening switches and the like without a problem). Often if an Ethernet cable is impossible, I connect the AP unit (usually a router configured as an AP) with a pair of high quality MIMO capable powerline adapters.
There are several types of bridges, but the one most pertinent for extending the range of wireless goes by a few different names. ASUS calls their routers set in this mode "media bridges." Using a high end pair of AC routers with one as the router and one as the media bridge is a reasonable solution to reach a distant point with relatively good bandwidth and reasonable latency.
DD-WRT has a similar setup called a client bridge that looks like
THIS to span a distance with wireless and connect wired devices to the hardware on the receiving end. There are other forms in DD-WRT like repeater bridges, which acts as a repeater and cuts your bandwidth so is suboptimal.
For minimum latency, a key for gaming, an AP is probably your best choice. You can have the AP close to the game unit and use either a wired or wireless connection to the AP. Wired to the AP will give you the lowest lag of any solution, and is about the same as simply connecting the gaming unit to your distant router with a great pair of powerline adapters.
THIS is the best source for real world information on powerline adapters in my opinion. I've tested many and always get very similar results.