Intuition behind homogeneous models I understand the formal definition of a homogeneous model but I have a hard time figuring out why it is an important property. In other words, I want to understand why we are interested in models that allow expanding partial elementary maps between subsets of their domains to automorphisms.
Along the same line, it would be really helpful if someone can give me an example of a countable model that is not homogeneous, so that I can better understand what homogeneity is about.
 A: For a model that is not homogeneous, consider the ordered set $\{1,2\}\times\mathbb Z$ with the lexicographical order.
All elements have the same first-order properties, so there's a partial elementary map that sends $(1,0)$ to $(2,0)$. But this doesn't extend to an automorphism.
A: Suppose $M$ is an $L$-structure, and we want to make sense of the idea that two elements $a$ and $a'$ "look the same" from the point of view of $M$. The strongest possible reasonable interpretataion of this notion is to say that there is an automorphism $\sigma$ of $M$ such that $\sigma(a) = a'$ (here reasonable means isomorphism invariant!). 
In particular, if there is an automorphism $\sigma$ of $M$ such that $\sigma(a) = a'$, then $\text{tp}(a) = \text{tp}(a')$. That is, the existence of the automorphism $\sigma$ verifies that the elements $a$ and $a'$ satisfy exactly the same set of first-order formulas in $M$. 
But the converse is not true in general. In a general $L$-structure $M$, is is possible that there are elements $a$ and $a'$ such that $\text{tp}(a) = \text{tp}(a')$, but there is no automorphism $\sigma$ such that $\sigma(a) = a'$. Such a situation is a weakness in first-order logic. That is, there are two elements that you can't tell apart using first-order formulas, but they don't entirely "look the same" from the point of view of $M$, in the strongest sense, since there is no automorphism of $M$ moving $a$ to $a'$.
But if $M$ is homogeneous, then $\text{tp}(a) = \text{tp}(a')$ if and only if there is an automorphism $\sigma$ of $M$ such that $\sigma(a) = \sigma(a')$. That is, this weakness in first-order logic goes away in homogeneous models. I hope this motivates why homogeneous models are useful in the model theory of first-order logic!
In fact, everything I've written above holds when $a$ and $a'$ are not just elements, but finite tuples. And when $M$ is a homogeneous models of cardinality $\kappa$, you can even allow these tuples to be infinite of length $<\kappa$.
