"maker of pots" (they also sometimes doubled as bell-founders), late Old English pottere "potter," reinforced by Old French potier "potter," both from the root of pot (n.1). As a surname from late 12c. An older Old English word for "potter" was crocwyrhta "crock-wright."
Potter's field "burying place for paupers, unknown persons, and criminals" (1520s) is Biblical, a ground where clay suitable for pottery was dug, later purchased by high priests of Jerusalem as a burying ground for strangers, criminals, and the poor (Matthew xxvii.7). The ancient Athenian city cemetery also was a "potterville" (Kerameikos). There seems to be an ancient pattern of association between potters' workshops and burial places (Argos, Rhodes, etc.; see John H. Oakley, "Athenian Potters and Painters," vol. III, 2014). Perhaps it was simply that both were kept away from the inhabited districts for the sake of public safety (disease on the one hand and on the other fires sparked by the kilns).