As far as I understood the question: 2-way Softmax means, the Neural Network should decide, weather it is this (x)or that. So it can only output a binary decision.
N-way means, that you have multiple output. For instance: Your network should decide, if a given picture shows a dog or a cat (2-way)
Another Network should tell, if the picture shows a car, a ball, a house or a human, etc... (n-way).
Check this out:
https://github.com/Kulbear/deep-learning-nano-foundation/wiki/ReLU-and-Softmax-Activation-Functions
Technically spoken: A bunch of neurons in a network are connected with mathematical operations, weights and biases...
Most models I use, use numbers between 0 and 1 (double/float). But in classification networks, you want to have one defined result. For the cat/dog network, you would have 2 output neurons (one for the cat-probability and another one for the dog-probability). So a given picture with a clear fat dog will produce something like (0.265, 0.995). So its most probably a dog. Applying the 2-way softmax here, will result in ~(0.2, 0.8). So "its a dog"
Basically: It normalizes the fuzzy output to enable the user to make a clear descision. Read the link provided. Its good...