A Buddha-Field is another word for a [Pure land][1], a realm associated with a Buddha. In the [Pure Land Buddhism][2] followers chant to a Buddha such as [Amitabha][3] in order to be reborn in his pure land where enlightenment is guaranteed thereafter. 

To be honest I'm not seeing much about Dharma-Fields when I look. But if you are seeing the term used in the same context as Buddha-Field I'm going to go out on a limb and say they mean the same. I'd be interested if someone else knows otherwise.

The question about [pure lands][4] give more information.

I hope that helps some.


  [1]: http://en.wikipedia.org/wiki/Pure_land
  [2]: http://en.wikipedia.org/wiki/Pure_Land_Buddhism
  [3]: http://en.wikipedia.org/wiki/Amit%C4%81bha
  [4]: http://buddhism.stackexchange.com/questions/1389/what-is-the-pure-land