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I've neglected to document this portion as well as I'd like.

For now, please just browse the source (it's not massive) and follow the types.

Callback

Included is a type CallbackTo[A] which captures effects designated for use in React callbacks. Callback is CallbackTo[Unit] with a different companion object, full of different goodies that all return Unit.
(See also USAGE.md.)

It is roughly equivalent to IO/Task in Scalaz, Haskell's IO monad, etc.

Living in the core module with no FP dependencies, many ops normally provided via typeclasses (eg. <*, >>=, etc.) are built-in directly. The Cats & Scalaz modules contain typeclass instances for it.

There's also CallbackOption which is a callback & option monad stack. Check out the online CallbackOption example.

Scalaz

libraryDependencies += "com.github.japgolly.scalajs-react" %%% "ext-scalaz72" % "1.0.1"

Included is a Scalaz module that facilitates a more functional and pure approach to React integration. This is achieved primarily via state and IO monads. Joyously, this approach makes obsolete the need for a "backend".

State modifications and setState callbacks are created via ReactS, which is conceptually WriterT[M, List[Callback], StateT[M, S, A]]. ReactS monads are applied via runState. Vanilla StateT monads (ie. without callbacks) can be lifted into ReactS via .liftR. Callbacks take the form of IO[Unit] and are hooked into HTML via ~~>, e.g. button(onclick ~~> T.runState(blah), "Click Me!").

Also included are runStateF methods which use a ChangeFilter typeclass to compare before and after states at the end of a state monad application, and optionally opt-out of a call to setState on a component.

There's only one example at the moment: State monad example.

Monocle

libraryDependencies ++= Seq(
    "com.github.japgolly.scalajs-react" %%% "ext-monocle" % "1.0.1",
    "com.github.julien-truffaut" %%%  "monocle-core"  % "1.4.0",
    "com.github.julien-truffaut" %%%  "monocle-macro" % "1.4.0"
)

addCompilerPlugin("org.scalamacros" % "paradise" % "2.1.0" cross CrossVersion.full)

A module with a extensions for Monocle also exists under ext-monocle.

There's one example online that demonstrates Monocle usage: ExternalVar example.

Cats

libraryDependencies += "com.github.japgolly.scalajs-react" %%% "ext-cats" % "1.0.1"

There's a Cats module now too. It's pretty much that same as the Scalaz module but without any IO stuff.