Our subjective experience of motion depends very much on our own motion as we observe. Consider for example a child playing ball in a car traveling along the freeway at constant velocity. An observer in the car would notice the projectile motion of the ball while an observer at the side of the road would mainly notice the motion of the ball along the freeway. The experience is different but the physical phenomenon is not. It is important that we be able to reconcile the observations of observers which move with respect to one another. Moreover we find that sometimes it can be useful to consider motion from conveniently chosen coordinate systems which facilitate the understanding of the motion.
To relate observations we write out the relationship between position
vectors of a particle P with respect to frames of reference A and B
Notice that 
is the position vector of the particle
with respect to the reference frame A etc. 
is the vector
from A to B, or the position vector of reference frame B with respect
to reference frame A. I recommend keeping symbols on vectors
as I have here. It is then easier to remember how the equation should look.
Differentiating this expression we obtain a relationship between velocities
of the particle measured in the two reference frames:
Finally with one last differentiation we relate accelerations:
Frames of reference which do not accelerate are called inertial frames.
As we see from the last two equations observers in inertial
frames measure the same acceleration of objects even though the measured
velocities are different and must be related to each other
through Eq. 18.
The simplest situation to deal with is when all velocities involved lie along a single direction then we are just dealing with scalar equations and the only place to go wrong is with the signs in Eq. 18. Lets look at a problem which is quite a bit harder than that:
A plane moves with air-speed 500 km/hr. When heading 20 degree east of north it moves 800 km due north in 2 hrs. What is the magnitude of the air velocity.
We first write the vector equation relating the various
velocities in the problem. For this we use our hand-rule for
indices:
We choose a coordinate system with 
pointing east and
pointing north. In this system we can have
and
Where we have introduced the angle 
which the direction of the
airspeed makes with north (). Inserting these expressions in
Eq. 21 we obtain
The magnitude of the velocity of the air with respect to ground is thus
One can also use geometry rather than coordinate representation here but
I am guessing that the coordinate representation will be easiest for most
of you to follow.
Take your time to get to know these problems. They are known to be hard to
understand.