Flat light means that there is almost no contrast within a photo, often it can give a new feel to an otherwise average photo. Low contrast can often be achieved in the editing process. Below is a a photo by William Wegman of his dog Man Ray. As you can see, this photo has very low contrast, it is quite grey all over, however, contrast still remains in the subject, the dog which is why this photo still works because the detail still remains.
High contrast means the blacks in the photo will be very black the whites very white, and very little grey in between. This is most noticeable in black and white photos.