Wireless Repeater / Extender vs. Access Point ?
Access point is a device connected with cable (Cat5) to your main router/modem/internet, and serving clients wirelessly.
Repeater is a wireless network device that repeats wireless signals to extend range without being connected with cable to either your router/modem, or your clients. Many routers/access points can be configured in "repeater" mode.
Extender has the same functionality as a repeater. Extenders are usually devices that only do repeating, and can't be set in any other mode.
The advantage to using a repeater/range extender is there is no need for a cable between the main router/modem and the repeater.
There is a downside to using a repeater vs. using an access point (or setting a device in repeater vs. access point mode), however - it essentially halves your wireless bandwidth, as it has to use it to communicate with both your main AP and the clients. Although range extenders/repeaters reach areas beyond the range of the central router, they also bounce back all the router's traffic, creating congestion and slowing the network.
It may be beneficial to use two access points (set on different, non-overlapping channels, with the same SSID/security) instead of range extenders/repeaters to extend wireless network coverage if a cable between your main router and the secondary access point is not an issue. Repeaters typically reduce wireless speed in half and introduce some noise.
If running an Ethernet cable is not feasible, one can use newer "mesh network" devices that use multiple radios, one for "dedicated backhaul" and others for serving clients.
See Also: How to configure a Wireless Router as an Access Point?