Why is the time interval between two events measured by two synchronised clocks seperated by a distance not proper? Suppose that in an inertial frame two events A and B occur at two different places. If all the clocks are synchronised then why is the time interval (difference between the time of event A registered by clock A and the time of event B registered by clock B) not proper time interval? Is it only because the readings are taken from two different clocks at different position?
 A: The proper time is the length of a world line between two points in spacetime. It is calculated using the metric, which in special relativity is the Minkowski metric:
$$ c^2d\tau^2 = c^2dt^2 - dx^2 - dy^2 - dz^2 $$
We often define a proper distance instead and this is related to the proper time by $ds^2 = -c^2d\tau^2$. Typically we use the proper time for timelike paths and the proper length for spacelike paths to avoid ending up having to square root a negative number.
The proper time depends on the world line. For any two points $A$ and $B$ there are an infinite number of paths that connect those two points, and those paths will in general have different proper times. However in introductory SR courses we frequently only consider straight lines joining the two points, and in that case for the two points:
$$\begin{align}
A &= (t, x, y, z) \\
B &= (t+\Delta t, x+\Delta x, y+\Delta y, z+\Delta z)
\end{align}$$
the proper time for a straight line joining $A$ and $B$ is simply:
$$ c^2\Delta\tau^2 = c^2\Delta t^2 - \Delta x^2 - \Delta y^2 - \Delta z^2 \tag{1} $$
Suppose in your example the two event are in the same place i.e. $A$ and $B$ are the same point. Then $\Delta x = \Delta y = \Delta z = 0$ so the proper time calculated using equation (1) is:
$$ \Delta\tau = \Delta t $$
So in this case the proper time is equal to the time interval between the events i.e. the proper time $\Delta \tau$ and the coordinate time $\Delta t$ are the same. But if the two events have a spatial separation the proper time and the coordinate time will be different.
The significance of the proper time is that it is an invariant i.e. all observers in all frames of reference will agree on the value of $\Delta \tau$. However different observers will disagree about the value of $\Delta t$.
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A: Proper time is your time measured by your clock. It is the time passed in your frame of reference regardless of how you moved. So the proper time is local to the observer. There is no such thing as "proper time between you and me" or "proper time at a distance" or "proper time between two separate objects".
A proper time interval is the distance between two events on a trajectory of an object in spacetime. If you have two objects, each would have its own trajectory and they would be separated. If you measure the distance between one event on one trajectory and another event on another trajectory, it is easy to imagine that the line connecting these points would not represent an actual spacetime trajectory of any object.
