Compensating euphoniums are for professionals or euphonium players that are really dedicated to playing. This is because compensating euphoniums are much more expensive than non-compensating euphoniums. All compensating means is that the instrument has the further advantage of being able to play in the lower octaves using conventional fingerings. See related link for more information.

This can be a matter of personal preference but normally the best brand of Euphonium is considered to be a Besson Euhonium. Other really good compensating brands are York, Willson, Sterling, and Yamaha.

compensating windings are used to overcome armature reaction

a compensating plate is used in michelson interferometer to compensate the additional path difference of ray 1

Compensating means you are providing something in place of something else. This could refer to compensating someone with money in exchange for their time, efforts, products or their work. Alternatively it can refer to someone perhaps doing favours for someone in place of contributing financially. It could also mean for example having a large expensive vehicle, but living in a park home (compensating for the lower standard of house with a higher standard of motor vehicle).