Trying to understand neural networks
Learning rate: that code does not use a learning rate, or rather it uses a learning rate of 1.  Lines 48,49 just add the adjustment (gradient) value without any rate.  That is an uncommon choice and can sometimes work in practice, though in general it is not advised.  Technically this is the simplest version of gradient descent, and there are many elaborations to simple gradient descent, most of which involve a learning rate.  I won't elaborate further as this comment answers the question that you are asking, but you should know that building an optimizer is a big area of research, and there are lots of ways to do this more elaborately or with more sophistication (in the hopes of better performance).
Error threshold: instead of stopping optimization when an error threshold is reached, this algorithm stops after a fixed number of iterations (60,000).  That is a common choice, particularly when using something like stochastic gradient descent (again, another big topic).  The basic choice here is valid though: instead of optimizing until a performance threshold is reached (error threshold), optimize until a computational budget is reached (60,000 iterations).
I'm new to coding and I've been guided to start with Python because it is good for a beginner and very versatile. I've been watching some tutorials online on how to  create a neural network with Python, however I've just got stuck in this example.
I've seen and worked out tutorials where you have the learning rate and the error threshold which are constant variables. For example learning rate = 0.1 and error threshold = 0.1, however in this particular example there are no constant learning rate and error threshold variables that I can see. 
Can someone explain why the learning rate and error threshold aren't being used?