The Arkendish can do many things, most of them connected to Thinkamancy. Charlie is a Carnymancer by trade, but has wielded the full powers of a Thinkamancer with the Arkendish's help.
With it, Charlie can handle an unlimited number of Thinkagrams, and extend some of his abilities to his personal Archons in his tower. Any commander in the world can get those Archons' attention by concentrating hard enough, for long enough. This could take hours or even days, but Charlescomm would establish contact eventually. This allows him to hire out to any side or barbarian in the world.
The signature unit of the Dish seems to be the Archons, flying female knight-class units with an array of powers. Charlie can pop one archon every turn or three archons every two turns with a turnamancer. Whether this capability is granted by the Dish is unknown. Most other cities cannot pop archons at all and those that can take multiple turns.
Extreme speculation: Mind control/influence, especially when directly communicating. (Maggie is very worried when Parson asks her to contact Charlie.)Erf-b1-p088
More extreme speculation: given that it can grant Thinkamancy to a caster of a different discipline, it might even be capable of enabling a four discipline link.
Real World References
The Arkendish resembles the antenna used for satellite television.
Charlie and his Archons are more than likely a reference to the American T.V. Show "Charlie's angels"
It was pointed out that Dish Network formed by Charles W. "Charlie" Ergen provides satellite television service in the United States.