Going into a bit more details regarding how things happen, there's a few levels:

Cores

At the hardware level the CPU is split into cores. It's just multiple processors linked together.

Processes

A core can run processes. Managing this is mostly the operating systems responsibility.

Threads

A process can start different execution threads. A thread is a "unit of work" that is processed in parallel. If the CPU has multiple cores, a thread will be executed on another core and be "truly parallel", otherwise If the CPU does not have multiple cores, it will alternate between the threads, processing a bit of each at a time.

Note: This is actually how the computer always runs. It switches between processes/threads all the time, but it happens so fast you don't notice it.