Downfield NMR refers to increasing chemical shift in the chemical spectra of a nucleus relative to the reference value. This is in contrast to upfield NMR, which refers to nuclei that have peaks which have decreased chemical shift.
The peaks in NMR spectra change depending on how shielded the nucleus is. More electrons results in more shielding, thus causing the spectra to shift upfield. If there are fewer electrons providing shielding than usual, the spectra shifts downfield. The chemical shift allows the bonds in molecules to be determined because the number of electrons available for shielding a particular nucleus depends on how many of the electrons are used in bonds with more electronegative nuclei, like chlorine or fluorine.