An ignoramus started out as a verdict, not a person. It was what a grand jury wrote when it refused to send a case to trial.
In Latin, ignoramus means "we do not know." It is the first-person plural of ignorare. In English law, a grand jury of 24 local men decided whether a charge should go forward. If they found the evidence too weak, the foreman wrote ignoramus on the bill. In practice, as Michael Quinion of World Wide Words puts it, it meant "we take no notice of this." English had the term by the 1570s, from Anglo-French legal use of the early 1400s.
The leap to "dunce" came from the stage. In 1615 King James I visited Cambridge, and the university put on entertainments. One was Ignoramus, a comedy mostly in Latin by George Ruggle, a fellow of Clare Hall. It was performed on 8 March 1615 before an audience Quinion puts at about 2,000. One contemporary grumbled that it was half spoiled by lasting six hours.
The title character was a common lawyer who spoke the clumsy "law Latin" that scholars despised. He was a caricature of Francis Brackyn, a lawyer and constant adversary of the university. James liked the play so much that he came back to see it again on 13 May.
The lawyers were less amused. Sir Edward Coke believed he was one of its targets. Lawyers answered with satirical poems and ballads, and the academics replied in kind. The quarrel made the character's name famous, and by 1616 "ignoramus" meant an ignorant person.
That history settles a grammar question. Some people write the plural as "ignorami," as if it were a Latin noun like "cactus." But ignoramus was never a Latin noun at all. It was a verb, so the plural is ignoramuses.
The word's family is the same as the verb it froze. Ignorant came into English in the late 1300s from ignorantem, the present participle of ignorare. Ignore arrived in the 1610s meaning "not to know." Its modern sense, "pay no attention to," is first recorded in 1801.