Understanding Feed Forward Neural Network Architectures

I have been reading through the architectures of Neural Networks and wanted to grasp the idea behind calculating the weights in a Neural Network and as you can see in the image below is a simple 2 node 2 layer Neural Network.

As you can see that for simplicity’s sake, I have just used a 2 node 2 layer network, but the same holds true for any sized Neural Network. It's just that the size of the Matrix increases proportionally to the number of inputs! The concepts outlined in the image holds gold!