Jackson is described as being thin with red hair. He has icy blue eyes, and apparently his nose resembles a knife. He is the brains to Jillson's brawn, though it's uncertain as to exactly how much of that he actually has.
It is unknown what connection he has to Jillson. He may be her brother, boyfriend, or fellow special recruits, as they are assumed to be in the first book. He terrorizes the Mysterious Benedict Society during their time at the Institute for the Very Enlightened. It was Jackson that yelled at S.Q. Pedalian for failing to recognize the spy outside the gymnasium. He was the Executive that retrieved Sticky and Reynie for their first Whisperer experience. He also spotted Kate during The Perilous Journey.
What happens to Jackson and Jillson in the first book may be a reference to the well-known nursery rhyme "Jack and Jill went up the hill to fetch a pail of water..." also because Kate drops her bucket, full of water, down on their heads in the first book.
In the Trivia book there is a spoof of Jack and Jill using these two.