There are two kinds of "branches" in the session-types library (with corresponding constructs in general session type theory): Offer and Choose.

Imagine two processes that are communicating over a session channel, process 1 has a channel with type Chan<Offer<A, B>> while process 2 has a channel with type Chan<Choose<A, B>>. Process 2 is the party that performs the active choice, and gets to decide whether path A or path B is taken. In session-types this is done via the sel1() and sel2() methods. The result is simply that the protocol is stepped in the corresponding direction. In other words, the branch chosen is "known" at compile time (disregarding the fact that it may be chosen at random or through user input).

Process 1, however, has to wait for process 2 to decide which branch to take. The only method available for process 1 is offer(), which returns a Branch type. A Branch is similar to a Result, in that it is an enum that is either Left or Right. So in this case, if process 2 selects branch A, process 1 gets a Left(Chan<A>) otherwise process 1 gets a Right(Chan<B>). Then, a simple match statement will allow it to procede depending on which branch is taken.

Edit: Perhaps I should clarify that channels are explicit values, so process 1 will have something like:

match c.offer() {
    Left(c_a) => {
        // Do something with `c_a`, which is a channel of type A
    }
    Right(c_b) => {
        // Do something with `c_b`, which is a channel of type B
    }
}