What does it mean that a core supports 2 threads? I understand that a process may have multiple threads, and that a processor may have multiple cores to run processes in parallel.

But I can't understand how a core may allow multiple threads, what does it change from a core with one thread support?
See Wikipedia on Computer Multitasking and Time-sharing, which have an explanation of how operating systems allow this.  Basically, they switch between multiple threads.

Read also about hardware multithreading.  This allows a single core to execute multiple threads concurrently, without switching between them.