Is fuel moved around during flight on a passenger or cargo plane? If so why?
Maybe

This is dependent on the air frame and varies from plane to plane and not all planes have capability of moving fuel although most large planes do. The Concorde moved fuel all over the place from its 13 tanks to cool the nose cone as well as trim the aircraft. It was perhaps one of the most complex implementations of such a system and kept the flight engineer quite busy.  



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Here is a layout of how it was used to control trim. 



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For any of this to be possible the aircraft must be equipped with cross-feed abilities. On larger planes this is fairly common, in case of a single side leak the cross feed valve is usually kept shut but since engines can have slight variances in fuel burn the pilots may need to cross feed every so often to balance the plane out. There is a nice overview on 737 fuel systems here.



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In small, often single engine aircraft fuel management is generally left up to the pilot. There are often multiple tanks which can be individually selected and must be rotated in flight every so often (30 min usually). Since some planes are prevented from drawing from "both" tanks, the switch has a lot to do with not running a tank dry in flight. 

This article overviews weight and balance fairly well. Often times quantity of fuel is more important than location. Most modern airliners store the fuel fairly close to the CG (in the wings or belly for that matter). The Concorde's unique delta wing design led to a fairly different use case than your run of the mill jet. 

Some of the balance issue is solved within the tank using baffles which keep the fuel from sloshing around too much but allow it to flow freely around a fairly large tank when need be.

You can find the full text of the FAA's book on fuel systems here.

Here is a neat video of how to actually cross feed fuel.