I don't understand the neural network. How does a perceptron work?

Question

Neural networks

I began to be interested in neural networks, and I wanted to repeat the perceptron algorithm, that is, the recognition of numbers from 0 to 3. I made a 25 by 25 field on the canvas so that I could draw. That is, as I understand it, I need 625 neurons and some weights. Well, in total, where there is a filled pixel, the input data will be 1 and everything else will be 0. well, then I make random weights from 0.0 to 1.0 for each neuron. and all this needs to be multiplied by the input data and summarized at the output. And as I understand it, I have 4 output neurons where the value is greater then and there, with a higher probability, the recognized number of the type. Well, I got some kind of game. It’s not clear how to find these weights, it’s all your life you need to sort through these weights as a password from something. I did not understand the meaning of hidden layers. And in general, I doubt that I correctly understood the perceptron

Answer 1

Take something faster. Why 600 neurons at once.
This should be more clear. Try to reproduce the same first. The source code for the example is available.
https://playground.tensorflow.org