Sufficiently well-behaved simple C*-algebras can be classified by means of a simple K-theoretical invariant.
In the non-simple case, the ideal structure must also be considered, and in the commutative case this tells the
whole story, but even in relatively well-behaved (non-commutative) cases the ideal structure intertwines
in a complicated way with the invariant referred to, unless this disappears completely, in which case the ideal
structure again determines the algebra.
Links:
[1] https://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/de/node/3444
[3] https://www.mpim-bonn.mpg.de/de/node/5312