Normalization is hard to visualize... We can all appreciate that.
This article (found here at the phpBuilder website) is a good start to beginning to understand how the normalization will apply in your case.
DB Normalization
The best thing to do, as this article illustrates, is to figure out what data can possibly be duplicated, and normalize from there...
(In my opinion, two manufacturers who manufacture a vehicle with the same name doesn't necessarily mean this becomes a many to many relationship, because as you research, you'll find that it is a rare enough instance, and they can be identified separately by one way or another. In this case, it is rare enough that it will be more of a headache to try to relate them in a many to many relationship) That's just my opinion though.
The first step is to understand what type of normalization you need, and you are heading in the right direction... Good luck with this.