What is 'energy compaction' in simple terms?
Yes, I believe that your understanding of energy compaction is correct.


  Does that mean, for any x, if A is DCT matrix, energy of y will be
  more concentrated when compared to the x.


No, it does not.  All that is needed to prove that such is not the case is to show that there is an $x$ that is not compacted by the DCT.  White noise, for example, would not be compacted by the DCT.  The DCT is useful because in many real life situations the "signal" (e.g. audio, images, videos, etc.) tend to be "pinkish", i.e. tend to have most of their energy in the lower frequencies and so there is a natural assymetry that can be used to our advantage.

Another way to look at this is from an information theory perspective.  If a signal is not "white" (i.e. it doesn't have the same power at all frequencies) that implies that there is some correlation between samples in the time domain, which means that the samples have "information" about the value of other samples.  This mutual information implies that there is redundancy, and thus should be able to reduce the amount of data without losing information.


  Does this happen to every x or x has to satisfy some properties to get this energy compaction?


For frequency-based transforms like the DCT it is clear from the above that what is needed is for the signal to be non-white.  The more non-white it is, the more compaction can be achieved.

There are other kinds of transforms though, which presumably could compact signals based on other features.  That is pretty much the whole point of compressed sensing.