What is the difference between a countable space and a discrete space? Are these the same concepts? 
I am particularly interested in space-time with a discrete structure and the like. I heard that there is not a standard way for quantizing space-time yet. At any rate, I would like also to know in a quantized space-time, coordinates merely take discrete values? Or they take discrete values only in some special quantum mechanical conditions?
 A: Defintion of a discrete set: click
Definition of a countable set: click
Finite sets are both, countable and discrete.
The set
$$ S := \{ \frac{1}{n} ~ | ~ n \in \mathbb{N}  \} $$
is countable and discrete.
The set
$$ S' = \overline{S} = S \cup  \{0\} $$
is countable but not discrete.
Proof: let f: $S' \rightarrow \mathbb(N) $ with $f(0) = 1$ and $f(1/n) = n+1$ then f is injective. Thus $S'$ is countable. However, $S'$ is closed and has an accumulation point $0 \in S'$. Thus it cannot be discrete (see definition).
A: I would understand discrete such that there is a minimum distance between the nearest element. A cubic lattice is an example of this.
In mathematics, the expression countable means that there exists a mapping from the natural numbers to the set you describe. This means that you can sequentially number every point. In a cubic lattice, this is certainly possible with an appropriate scheme. So a cubic lattice (as in an ordinary solid crystal) would serve as a countable and discrete space.
The rational numbers $\mathbb Q$ are also countable. Yet I would not call it discrete because you can always find (countable) infinitely many numbers between any two numbers. If you would make a spacetime which was $\mathbb Q^{1,3}$ instead of the usual $\mathbb R^{1,3}$ it would be countable but I wouldn't call it discrete.
A discrete spacetime is used in K. Wilson's lattice field theory where it serves as a regulator. The coordinates there do take discrete values, you just number the lattice sites with numbers from $\mathbb N^4$.
In quantum field theory, spacetime is not quantized. It is also uncountably infinite.
About the “no standard way”: Discretizing the space with a lattice is a computation aid. It has no physical interpretation. It is just darn useful when doing simulations on a computer!
There are thoughts that spacetime might be discrete in a physical way. One of this is quantum loop gravity which discretizes space at the order of $10^{-34} \, \mathrm m$. This is so small that nobody has any idea what could happen there.
A: "Countable" and "discrete" are separate concepts. The Cantor set, for example, is discrete, but not countable.
As for discretizing space-time, the only theory I'm aware of that takes this approach is known as loop quantum gravity. String theory, if my passing understanding is correct, does not discretize space-time in this way. 
