There doesn't have to be any relation. A computer can have any number of cores; a process can have any number of threads.
There are several different reasons that processes utilize threading, including:
Programming abstraction. Dividing up work and assigning each division to a unit of execution (a thread) is a natural approach to many problems. Programming patterns that utilize this approach include thereactor,thread-per-connection, andthread poolpatterns. Some, however, view threads as an anti-pattern. The inimitable Alan Cox summed this up well with the quote, "threads are for people who can't program state machines."
Blocking I/O. Without threads, blocking I/O halts the whole process. This can be detrimental to both throughput and latency. In a multithreaded process, individual threads may block, waiting on I/O, while other threads make forward progress. Blocking I/O via threads is thus an alternative to asynchronous & non-blocking I/O.
Memory savings. Threads provide an efficient way to share memory yet utilize multiple units of execution. In this manner they are an alternative to multiple processes.
Parallelism. In machines with multiple processors, threads provide an efficient way to achievetrue parallelism. As each thread receives its own virtualized processor and is an independently schedulable entity, multiple threads may run on multiple processors at the same time, improving a system's throughput. To the extent that threads are used to achieve parallelism—that is, there are no more threads than processors—the "threads are for people who can't program state machines" quote does not apply.
The first three bullets utilize threads with no relationship to cores. If you are using threads as a programming abstraction to handle UI elements, for example, you'll have one thread per UI element (or whatever) regardless of whether you have 1 core or 12. Similarly, if you were using threads to perform blocking I/O, you'd scale your thread count with your I/O capacity, not your processing power.
The fourth bullet, however, does relate threads to cores.If the goal of threading is parallelism, then the number of threads should scale linearly with the number of cores. For example, if you double the number of cores in a system, then you would double the number of threads in your application. This is true for cores in the logical sense—that is, including SMT.
When threading is used to achieve parallelism—and this is both a common and the best use of threading—you will often have, say, one or two threads per core. Oftentimes, applications are written so as to dynamically size thread pools off the number of available cores. A single thread per core is ideal, but applications often use a larger multiplier, such as two threads per core, due to bugs and inefficiencies in their code, such as operations that block when none should.
What is the relation between the number of threads and the number of CPU cores?