Neural networks consists of neurons, connections between these neurons called weights and some biases connected to each neuron. We distinguish between input, hidden and output layers, where we hope each layer helps us towards solving our problem. To move forward through the network, called a forward pass, we iteratively use a formula to calculate each neuron in the next layer. Keep a total disregard for the notation here, but we call neurons for activations $a$, weights w and biases b-- which is cumulated in vectors. This takes us forward, until we get an output.

Mathematically, this is why we need to understand partial derivatives, since they allow us to compute the relationship between components of the neural network and the cost function. And as should be obvious, we want to minimize the cost function. When we know what affects it, we can effectively change the relevant weights and biases to minimize the cost function. If you are not a math student or have not studied calculus, this is not at all clear. So let me try to make it more clear. The squished'd' is the partial derivative sign.