Also, what's the difference between transforming a problem into another problem and doing a reduction? They sound synonymous to me. Thank you!
To prove that a problem is NP-complete, you need to show two things:
- The problem is NP-hard, i.e. it's at least as hard as any problem in NP
- The problem is in NP, i.e. it's no harder than NP
Doing a reduction both ways is the easiest way to do this: reducing something in NP to a new problem shows that the new problem is NP-hard, and reducing your new problem to something in NP shows that the new problem is no harder than NP.
I've never heard "transforming" used formally, but I would guess from context it's the same as doing a polynomial-time reduction.