The phrasing I am confused about is "thus the function $f$ may be finite over a point". What exactly is meant by this, and why does it follow from the previous sentence about $f$ being integral? Does this mean finite as a morphism of schemes? A morphism to where though, the spectrum of the residue field?
For reference, this is from page 139 of Mumford & Oda. A draft electronic copy is available here (on Mumford's web page) in which the remark appears on page 109.