Will this run new titles ok at 1280x1024

Ethernet is running an ethernet cable directly from a device such as a pc to the router/modem which allows internet access. The cable is preferred because you always have access to internet until you unplug the ethernet cable or the internet cuts out.
Wifi transmit radio signals at a certain frequency range. You need a wifi router and wifi adapters or wifi compatible devices. The wifi adapter can convert those radio signal into electrical signals and you have access to internet. Whilst it is wireless, it is not a sustained signal. there can be object which reduce the strength of the signal hence your internet slows or cuts out for a few seconds.
This is a good video at visual explaining as well: https://www.youtube.com/watch?v=Sue1Zvmh8JA
 


Yes it will work since it is a wifi router. You also have a wifi password stated on the router by a sticker which you will need. It will also have a name, make sure you know which wifi network is yours.