Chair: (Here) a Sedan Chair - a covered chair supported by poles, carried by two bearers.
Episcopal: To do with (here appointed by) a bishop - the adjective refers to church administration at the time Swift wrote.
Gibbet: Place where criminals are hanged.
Mandarin: Important official serving an oriental (originally Chinese) ruler, or any high official today.
Papists: Supporters of the Pope, an insulting name for Catholics.
Pretender: James Stuart, a Catholic who pretended to (claimed) the English and Scottish thrones. He is sometimes known as the Old Pretender, while his son, Charles Edward Stuart, is known as the Young Pretender (or Bonnie Prince Charlie)
Shambles: Place (usually in a town) where animals are slaughtered and butchered.
Solar year: A year in the ordinary sense (as measured by the earth's going once round the sun).