Definitions

space time:

spacetime is the frane where an event occurs. That is where 
the position of a point is given by its coordinates in 
space and time. t

event:

An event is an occurrence localized in space and time.

frame of reference:

A frame of reference refers to a coordinate system 
within we make measurements.

inertial frame:

an inertial frame of reference, or Galilean reference frame, 
is a frame of reference that is at rest or in rectilinear motion at 
constant velocity with respect to another frame, then not accelerated.
In this frame, time and space are homogeneous (the same) 
and isotropic (uniform in all directions).
Measurements in one inertial frame can be converted to 
measurements in another inertial frame , by Galilean transformation 
in Newtonian physics and the Lorentz transformation in special 
relativity.

observer:

The term "observer" refers to hwo is making local measurements

proper length:

The proper length of an object is the length of this object 
as measured by an observer which is at rest relative to the 
object.

proper length between two events:

The proper length between two space-separated events is 
the distance between these two events when they are 
simultaneous. 

In any inertial frame of reference, the proper 
length L is: 
L = [Δx2 + Δy2 + Δz2 - c2Δt2]1/2
Where:
-- Δx,  Δy,  and Δz are the difference of spacial coordinates 
of the two events.
-- Δt is the difference in the temporal coordinates of the two events,
-- c is the speed of light.

simultaneity::

Simultaneity is the fact that two events occur at the same time, 
unlike of absolute. Simultaneity  depends on the observer's 
reference frame.

Lorentz invariant :

In a Euclidean space, the separation between two points 
is the the distance between the two points. this distance is
spatial and positive. In spacetime, the separation between 
two events is the interval  between the two events, that 
takes into account both the spatial and tempora separation 
between the events. The spacetime interval is:

s 2  = r2  - c2 t2 

priciples of relativity:

-- The laws of Physics take the same form in all inertial frames 
of reference.
-- The speed of light in vacuum is the same in all inertial frames 
of reference. It is independent of the motion of the source or 
the motion of the receiver.