Blazon: Per quarterly I and IV argent on a cross azure five crescents or (Piccolomini); II and III paly of four i or four palets gules (Aragon), ii barry of eight gules and argent (Hungary ancien), iii azure semé de lis or, a label of four points gules (Anjou ancien), iv argent a cross paté between four crosses or (Jerusalem)
Ottavio inherited the quarters of Aragon, Hungary ancien, Anjou ancien, and Jerusalem from his ancestor Antonio Piccolomini d’Aragona, who married Maria d’Aragona, the illegitimate daughter of Ferdinand I of Naples. Many representations have Ferdinand’s arms in the first and third quarters, as Maria’s lineage was (though illegitimate) more noble than Antonio’s.
Let me say this, that this coat-armor being Barry, is interpreted by some learned in blazon, to represent to the bearer, force, valour, courage, or wisdom, whereby he hath repelled any peril or danger imminent to his country or sovereign; the Barre… is taken in the like signification.
– From the Blazon of Gentrie by Sir John Ferne (1586) p243
Like most early attempts to assign some kind of symbolism to heraldic charges (especially the ordinaries), this assertion does not seem to be supported by more reliable sources.
Blazon: Per pale or a bull passant gules on a base vert within a bordure of the field charged with eight flames of the last and barry of six or and gules*
Rodrigo used these arms at least since his election to the papacy; he may have taken that opportunity to add to his family’s traditional arms. I cannot find any record of the origin of the sinister arms, though several other later Borgias used them as well. They may be a reference to the arms of Aragon, where he was born.
Blazon: Vert a pile barry argent and sable surmounted by a chief per fess dancetty of the second and the first
Supposedly, the arms represent highly stylized geographic features; the three points of the dancetty partition line being mountains, and the pile on the field vert being a dam in the steep valley where the town is located.
Blazon: Per quarterly I and IV lozengy argent and gules (Appiani), II and III per quarterly i and iv tierced per pale 1 barry of eight gules and argent (Hungary ancien), 2 azure semé de lis or (France ancien), and 3 argent a cross paté between four crosses or (Jerusalem), ii and iii or four palets gules (Aragon); overall a chief gules a cross argent (Savoy)
“Louis” appears to be Alfonso d’Appiani, brother of Jacopo VI. He seems to have been a military man, possibly serving under Ferdinando II de’ Medici, Grand Duke of Tuscany.