Can one core have four threads?

I'm curious to understand the intricacies of modern computer architectures. Specifically, I've heard that some processors can have multiple threads per core. But does this mean that a single core can indeed handle four separate threads simultaneously? Wouldn't that require more than one execution unit within the core? How does the operating system and processor management software ensure that these threads are efficiently scheduled and executed? Is there a significant performance difference between a true multi-core setup and a single core with multiple threads? What are the trade-offs to consider when deciding between the two?