Access Point or a Repeater ?

Access point is a device connected with network cable to your main router/modem/internet, and serving clients wirelessly.
Repeaters/Extenders are devices that repeat wireless signals to extend range without any need for being connected with a network cable to either your router/modem, or your clients. Some routers/access points can be configured in "repeater" mode as well.

The advantage to using a repeater/range extender is there is no need to run a network cable to the repeater itself, it can connect wirelessly to both your network/internet, and extend that wireless connection to your clients. There is a downside, however - it sessentially 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.

The advantages to using an access point vs. a repeater/extender is that it is much faster, as it only needs to communicate wirelessly with clients, reducing load on the unit and wireless congestion/collisions. The one disadvantage being that you have to run a Cat5 network cable to the access point.

Note you can have multiple access points setup on different, non-overlaping channels, with the same SSID/security and your clients should be able to switch between them and automatically connect to the one with the strongest signal.