A widget is something that you can play on the screen of your phone that shows you some sort of information or function. There are numerous widgets available on the various android phones. There are weather widgets that can use the GPS on your phone. They give you the current weather conditions, forecasts etc. There are widgets that can give you news updates, act as switches for wifi, bluetooth, gps, and etc.
Those are a few things that come to mind off of the top of my head. Applications are various programs you can access, but aren't necessarily running in the background the entire time. the phone is powered on.