Are there any examples of two-valued logics that are not classical? What are some examples of nonclassical two-valued logics? How would such a logic work? What relationship do the non-classical two-valued logics have to ordinary two-valued logic?
This answer includes an interesting parenthetical comment.

Also, as a fairly subtle point, you can have a $2$-valued logic that isn't classical.

The question under consideration is specifically about propositional logic, so I'm assuming that this answer is also talking about different propositional logics.
The only things I can really think of that might qualify as non-classical two-valued logic are simple, somewhat bizarre systems like a fragment of classical logic where the only connective is $\land$, or one where the only connective is $\to$. Neither example system has truth value constants $\top$ and $\bot$. I'm not at all convinced that these examples are really two-valued, or are not better analyzed with different, non-two-valued, semantics.
 A: First of all, fragments of classical propositional logic gotten by reducing the language can actually be rather interesting. So those examples aren't unnatural. (They're discussed IIRC in Blok/Pigozzi's text which I'll cite more centrally below.)
That said, here's one you may find more compelling, since it involves $(i)$ an increase in strength and $(ii)$ a new "type" of logical operation. Let $\mathcal{L}$ be the usual language of propositional modal logic (so we have a set $P$ of propositional atoms, the usual classical connectives, and the new unary symbols $\Box$ and $\Diamond$). Equip $\mathcal{L}$ with the "pointed Kripke frames" semantics:

A $\mathcal{L}$-structure is a tuple $(W, R, w, \nu)$ where:

*

*$W$ is a set of things called worlds and $R\subseteq W^2$ is any relation;


*$w\in W$ is the distinguished world; and


*$\nu$ is a valuation map from $P$ to $\mathcal{P}(W)$.

The "models" relation $\models_\mathcal{L}$ is based off the usual Kripke semantics: we say $(W,R,w,\nu)\models\varphi$ iff $\varphi$ is made true at world $w$ in the Kripke frame $(W,R)$ by the valuation $\nu$.
Now this is arguably two-valued: for every $\mathcal{L}$-structure $\mathfrak{A}$ and every $\mathcal{L}$-sentence $\varphi$, exactly one of $\mathfrak{A}\models_\mathcal{L}\varphi$ or $\mathfrak{A}\models_\mathcal{L}\neg\varphi$ holds. However, it's definitely not classical. Specifically, the operations $\Box$ and $\Diamond$  - thought of as functions on the set of sentences - are non-truth-functional, and that's not something which can occur in classical logic. Moreover, this isn't a silly example at all; Kripke frames with a distinguished world are definitely reasonable things to consider.

OK, but now we come to a follow-up:

I'm not at all convinced that these examples are really two-valued, or are better analyzed with different, non-two-valued, semantics.

In particular, one might object that the non-classicality of the "pointed frame logic" above, stemming from the non-truth-functionality of its syntax, is actually an indictment of its claimed two-valued nature. Presumably this objection would go something like this:

A "truth value" is supposed to be a total description of the relationship a given sentence has to truth. When we have a putative notion of truth value with respect to which our linguistic constructions are not truth functional, this is a sign that that putative notion is not capturing all the truth-related data of the sentences in our language.

(Weirdly I don't have a citation for this at the moment, although it's an obvious line of argument; I'll add one when I can find it. FWIW I believe this is not generally held amongst non-classical logicians, although I personally have a lot of sympathy for it.)
This whole line of thought takes us into rather odd territory: is there even a meaningful way to compare different semantics (semanticses? semantices? .... semantixoi!) for a given logic, or to talk about which semantics is "right?"
It turns out that there is actually a lot to be said about this! This is one of the topics treated in algebraic logic, and specifically in the context of (proto-)algebraizability of a logic. Blok/Pigozzi is a great introduction to this topic - it's well-written, full of examples, and extremely short. Nonclassical logic for the bus ride, indeed!
(And if by contrast what you want is a new dining room table, I can't not recommend Humberstone's book. Fun fact: you can use BP as a bookmark in H.)
If I recall correctly, if we interpret things strictly enough you can indeed prove that classical propositional logic is the strongest two-valued propositional logic in a certain sense. But care is always required with such results (e.g. we have to rule out things like infinitary conjunction), so I'm hesitant to outright claim it without a citation ... which I can't find at the moment. Regardless, the "algebraization" context provides a framework for understanding this type of question which I think will prove valuable to you.
A: Here's a simple two-valued logic that is nevertheless substantially different from classical logic.
Let's consider classical two-valued logic with the $\land$ and $\lnot$ as connectives.
Here are their truth tables. I'm writing the connectives above the truth table because in my browser ∧ is not the same width as a digit.
∧
    0 1
0 | 0 0
1 | 0 1

And for negation.
¬
0 | 1
1 | 0

The standard consequence relation $\models_{\mathsf{CL}}$ can be defined as follows.
$$ \Gamma \models_{\mathsf{CL}} \varphi \;\;\textit{if and only if} \;\; \text{for all valuations $v$, if $v \models \Gamma$, then $v \models \varphi$} $$
We can also define a new, non-vacuous consequence relation as follows.
$$ \Gamma \models_{\mathsf{NV}} \varphi \;\;\textit{if and only if}\;\; \text{$\Gamma \models_{\mathsf{CL}} \varphi$ and there exists a valuation $v$ such that $v \models \Gamma$} $$
Let $L$ be the set of well-formed formulas in propositional logic.
$(L, \models_{\mathsf{NV}})$ and $(L, \models_{\mathsf{CL}})$ have difference consequence relations and are therefore different logics, but they are both clearly two-valued.
As a quick demonstration that they are different:

*

*$A, \lnot A \models_{\mathsf{CL}} B$ is valid.

*$A, \lnot A \models_{\mathsf{NV}} B$ is not valid.

I saw this type of consequence relation in this lecture. Hitoshi Omori is referring a system that was, I think, originally devised by Graham Priest. It is discussed at around 20:25.
