Why do Newton's laws hold in a normal and tangential coordinate system? In my book it says:

When applying the equations of motions, it is important that the acceleration of a particle be measured with respect to a reference frame that is either fixed or translates  with a constant velocity. 

A n-t system follows the path of the particle, so if the path is bend (ergo there will be a acceleration in the normal direction) and the particle's velocity increases (ergo acceleration in tangential direction), how can it then be used for evaluating using the laws of newton?
The system follows the particle so to keep up if the particle is increasing in velocity so must the n-t system?
Therefore there is an accelerating frame of reference and it is not inertial frame of reference?
Which concept am I missing?
 A: I think confusion of the OP is related to the concept of the frame of reference. Frame of reference is a position in the space not a coordinate system. In other word, frame of reference is the position that measurements are evaluated with respect to it and for this evaluation, one can use any kind of coordinate system. So, the book is right when says:

When applying the equations of motions, it is important that the acceleration of a particle be measured with respect to a reference frame that is either fixed or translates with a constant velocity

The book doesn't refer to the coordinate system. It just refers to the reference frame.
Consider to figure below. In the figure I have chosen point $O$ as my frame of reference i.e. I measure what I want with respect to the $O$. The book says we can apply equations of motion if $O$ is fixed or translates with a constant velocity. But for applying the equations of motion, we can use any kind of coordinate system. I have showed two kinds of coordinate system in the figure.

As you can see, point $O$ (frame of reference) has not been changed for both cases.

It is useful to mention that in a Cartesian coordinate system ($x-y$), configuration of unit vectors is parallel for all positions of the particle but configuration of unit vectors in path coordinate system ($n-t$) can vary just like configuration of unit vectors in a polar coordinate system ($r-\theta$). You can see this in figure below.

A: What your book meant was, "When applying the equations of motion, it is important that the acceleration of a particle be measured by an observer traveling in a reference frame that is either fixed (in space) or translates with a constant velocity'.  The n-t coordinate system is fixed in space, and is not traveling along with the particle.  The reference frame of an observer locked to this coordinate system is fixed in space.  
