What does it mean to divide space and time? Goldstein's mechanics book, on the chapter on relativistic mechanics says that "We cannot assume that all observers make the same division into time and space in the same way." What does it mean to "divide" spacetime?
 A: Spacetime $M$ is made of elementary objects called events and the set of events can be equipped with four coordinates (you need four numbers to fix an event). However there are many ways to define systems of coordinates, and they do not distinguish necessarily space form time. 
The rest physical $3$-space of a reference frame or "observer", is obtained by fixing a class of $3$-surfaces $\Sigma_t\subset M$ pairwise disjoint, whose union is the whole spacetime $M$. They are labeled by the parameter $t$: it is the time coordinate of the considered observer.
Two events $a$ and $b$ happen at the same time $t_0$ for that observer if (and only if) they belong to the same $3$-surface: $a,b \in \Sigma_{t_0}$.
Changing observer, the events are not changed but the class of $3$-surfaces,  $\Sigma'_\tau$, describing the rest spaces of that observer  are different.
So it may happen that t $a,b \in \Sigma_{t_0}$, but there is no $\tau$ such  that 
$a,b \in \Sigma'_{\tau_0}$. The events $a$, $b$ happen at different time for the second observer. 
In general, if a subset $A$ of events can be completely included in a rest space for the first observer i.e, $A\subset \Sigma_{t}$ for some $t$, we can think of $A$ as a portion of  space for that observer. However, in general, $A$ cannot be completely included in any rest space of the other observer: $A\not\subset \Sigma'_{\tau'}$ for every $\tau'$. So, what for the former observer is space, it
is in general a mixing of space and time for the latter. 
The statement:
"We cannot assume that all observers make the same division into time and space in the same way" should be more clear, now.
A: I think the concept is best explained with an analogy.
We usually think of ourselves as existing in some three-dimensional space that we can move around in. We can move forwards and backwards, left and right, and up and down. However, what I perceive as forwards and backwards, somebody else, looking at things from another direction, might perceive as left and right (that is, somebody watching me walk across the room might see me walking from left to right. But I would perceive myself as walking forwards). I could place down some co-ordinate system in front of me, that declared 'up' to be pointing towards the sky, 'forwards' to be pointing towards the laptop I'm looking at, and 'left' to be pointing towards my left.
But not all people would divide up space in the same way. Somebody off on some space station looking at me could place down some equally valid co-ordinate system. It would be related to mine in such a way as this (in 2D for simplicity):
$$ x' = x \cos \theta - y \sin \theta $$
$$ y' = y \cos \theta + x \sin \theta $$
where $(x',y')$ are his co-ordinates, $(x,y)$ are mine, and $\theta$ is the angle between the directions we are facing. From his perspective (which is no more or less legitimate than mine), $x$ and $y$ have been mixed up --- that is, his co-ordinates are in some sense mixtures of mine.
Suppose we move to 3D, and introduce a $z$ co-ordinate. In just the same way that there is no 'natural' division of $x$ and $y$ (which we could take to mean 'forwards' and 'left' respectively), so the $z$ co-ordinate is not in any sense special or separate from the others. All it takes is a small rotation about the $x$-axis (say), and we will have mixed some of $y$ and $z$ together. What I would think of as 'up', you would think of as 'partly up and partly left'.
It's a very profound fact about the universe that time $t$ behaves (pretty much) just like $z$ --- it's not separate from $x$ or $y$ or $z$ in any natural way. Just as somebody rotated with respect to me assigns his $x$ and $y$ and $z$ differently to me, so somebody moving with respect to me will assign his $x$ and $y$ and $z$ and $t$ differently to me.
In one spatial dimension, the transformations look like this
$$ x' = \gamma\, x - \gamma \beta\, ct $$
$$ ct' = \gamma\, ct - \gamma \beta\, x $$
where $\beta = v/c$ and $\gamma$ is the Lorentz factor, and where $v$ is our relative speed. We can write these in an even more compelling way by defining $\tanh \phi = \beta$. Then the above equations become
$$ x' = \cosh \phi \, x - \sinh \phi \,ct $$
$$ ct' = \cosh \phi \, ct - \sinh \phi \, x$$
So to you, moving relative to me, my time and space have been mixed up. They cannot be consistently separated in just the same way that $x$ and $y$ cannot be consistently separated. What is meant by forwards, or left, or up, is just a matter of perspective (more precisely, of relative orientation). And similarly, what is meant by time and space is also just a matter of perspective --- it depends on relative velocity.
So it turns out that we don't exist in some three-dimensional space. We exist in a four-dimensional 'space', known as spacetime. When we say that different observers will 'divide up' spacetime differently, we mean that different observers will choose their co-ordinates $(x,y,z,t)$ differently, where the inclusion of $t$ here is crucial! It means that different observers will choose their 'time direction' differently!
Hope this helps.
