"Beautiful" has several meanings. Some people think clever code is 
beautiful. Some people think code that reads like poetry or prose is 
beautiful. Some people think codes that achieve a lot in a few 
characters is beautiful. Some people think code that respects an 
arbitrary paradigm (say Object-Oriented Programming) is beautiful.

Like with other arts, beauty is a matter of taste. A piece of code could 
be ugly for you and beautiful for me, or vice versa. Moreover, even for 
a single person, beauty is hard to define.

My definition of beauty is close to that of simplicity. This answer 
doesn't mean much since I cannot define simplicity precisely either.

David Gelenter is one of the Computer Scientists who thought the most 
about the aesthetics of code. He wrote:

  Beauty is more important in computing than anywhere else in technology
  because software is so complicated. Beauty is the ultimate defence
  against complexity.

More pragmatically, here are four important characteristics of beautiful 
code:

- de-coupling: code is structured in self-contained, re-usable, 
independent chunks;

- simplicity of interface: these code chunks do not take too many 
arguments and their behavior does not change too much depending on the 
arguments they take;

- evolutivity: adapting the code to new constraints (this includes 
business domain changes as well as portability) does not require major 
changes in the design of the software;

- coherence: if similar problems are solved in different parts of the 
software, they are solved in similar ways - opinionated software is 
usually coherent.