How does time dilation actually work in relativity?

[EvelynR]

I've been studying relativity and the concept of time dilation keeps coming up. I understand that time passes differently depending on your speed and gravitational field, but I'm struggling with the practical implications. If someone travels at near light speed, they age slower than people on Earth. But how does this actually work from a physics perspective? What's the difference between special relativity time dilation and general relativity time dilation? And what about the twin paradox - how is that resolved?