What is this thing on the engine of a Cessna 500 Citation I? I've noticed something on the engine of this bizjet. Could somebody explain what that is? 


Source (image modified to indicate location of fairing).
Those are the fairings containing the mechanisms for the engine's reverse thrust system. See this photo (copyrighted) of the similar reverse thrust "buckets" deployed on a Citation 550.

See this example of a similar thrust reverser system deployed on a different aircraft type:



Image source, public domain

See also Wikipedia's article on Thrust Reversal systems.