When giving blood, what is the specific Iron measurement they take?
Most likely they were measuring the hemoglobin concentration, in units of g/dL (i.e. 101 kg/m3). Assuming by your username you are male, the reference range for [Hb] is approximately 13.8 to 18.0 g/dL (sources vary somewhat), which roughly matches the "19" you were told was the upper bound.

Wikipedia also notes that [Hb] "is typically tested before or after blood donation", making it likely that that was what they measured.

Calling it "iron" isn't strictly correct. The human Hb tetramer's molecular weight is about 65 kDa, yet it contains only 4 iron atoms, so the actual concentration of iron in your blood is actually much less.

Useful resource: Wikipedia's list of reference ranges for blood tests has a very detailed chart of commonly cited reference ranges for many substances, with citations.