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That second piece is a program, written by humans, that defines the shape of the network. How many layers. How many neurons in each. Which neuron connects to which. It doesn't change during training and it doesn't change after.
This is worth slowing down on, because it's where two ideas can blur together. The neurons and their connections are the structure. The weights are the numbers that sit on those connections, one weight for each link, setting how strongly a signal passes along it. So the program lays out the empty wiring; the weights are what fill it in. The structure tells you how to read all those numbers, and which number belongs to which connection.
So every model is really two things (a slight simplification, but a useful one). The program is small, often just a few hundred lines of code, published openly for anyone to read. The weights are enormous, tens or hundreds of gigabytes of numbers, sometimes split across dozens of separate files.
When you run a model, the program loads the weights and uses them. The program is the shape. The weights are everything learned. Neither works without the other.