What are examples of local rings with a nontrivial prime ideal? I've just learned about the idea of local ring, but the only concrete examples in the book are $\{p/q \in \mathbb{Q} \mid q \mbox{ odd}\}$ and the power series ring $k[[x]]$ for some field $k$. But the only prime ideals they have are just $\{0\}$ and their maximal ideal.

Can anybody provide me with more concrete examples of local rings with "non-trivial" prime ideals?
First consider the polynomial ring $R=F[x,y]$ for a field $F$. One chain of prime ideals in this ring is $(0)\subseteq (x)\subseteq (x,y)$. Now $(x,y)$ is a maximal ideal, but there are other maximal ideals, for example $(x+1,y)$.

The easiest way to eliminate the other maximal ideals is to pass to the localization at the prime $M=(x,y)$ so that the new ring $R_M$ is a local ring. It is a property of localization that the prime ideals contained in $M$ will have prime counterparts in $R_M$, with the same containment relations and everything. Thus the chain $(0)_M\subseteq (x)_M\subseteq (x,y)_M$ will be a properly ascending chain of prime ideals in the new ring $R_M$.

Even if you are not handy with localization now, you probably will need to be soon, so trying to understand this type of example is worthwhile.
You will soon learn about localization and that will provide many many examples.

An example: Let $R$ be the ring of rational functions $f/g\in\mathbb Q(x,y)$ such that, when expressed in irreducible terms, the denominator is not divisible by $x$.