A Buddha-Field is another word for a Pure land, a realm associated with a Buddha. In the Pure Land Buddhism followers chant to a Buddha such as Amitabha 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 landspure lands give more information.
I hope that helps some.