This is a real neural network, and it is training in your browser right now. There is no library behind it, only about eighty lines of arithmetic. Press Train and watch it find the shape of the data.
Each dot is a training example. The colour behind it is the network's current opinion.
Each point goes in, and numbers flow through the weights to give an answer between 0 and 1. At first the weights are random, so the answers are nonsense.
The loss is a single number for how wrong all those guesses were. Everything the network does next exists to push that number down.
Backpropagation works out how much each weight contributed to the error. Each one is nudged a tiny amount in the helpful direction. Then it all happens again.
Pick Spiral and set one layer with two neurons. It cannot draw that shape, however long you wait, because it does not have enough neurons. Now set the learning rate to the maximum on Circle and watch the loss jump about, because each step overshoots the valley it is trying to reach.
Tick things off as you go. Your progress is kept in this browser.