What is the difference between (objective / error / criterion / cost / loss) function in the context of neural networks?
The error function is the function representing the difference between the values computed by your model and the real values. In the optimization field often they speak about two phases: a training phase in which the model is set, and a test phase in which the model tests its behaviour against the real values of output. In the training phase the error is necessary to improve the model, while in the test phase the error is useful to check if the model works properly.

The objective function is the function you want to maximize or minimize. When they call it "cost function" (again, it's the objective function) it's because they want to only minimize it. I see the cost function and the objective function as the same thing seen from slightly different perspectives. 

The "criterion" is usually the rule for stopping the algorithm you're using. Suppose you want that your model find the minimum of an objective function, in real experiences it is often hard to find the exact minimum and the algorithm could continuing to work for a very long time. In that case you could accept to stop it "near" to the optimum with a particular stopping criterion.

I hope I gave you a correct idea of these topics.