My first neural network made by following the book "Make Your Own Neural Network: A Gentle Journey Through the Mathematics of Neural Networks, and Making Your Own Using the Python Computer Language" by Tariq Rashid.
There are two examples:
One is simple XOR problem which is inspired by Daniel Shiffman from "The Coding Train" YouTube channel.
Link to the "The Coding Train" video: https://www.youtube.com/watch?v=188B6k_F9jU
Second example is recognizing handwritten number using MNIST handwritten digit database.
Existing datasets are smaller due to size limits. Full version .csv files: https://pjreddie.com/projects/mnist-in-csv/