There are a few different types of cost functions, but for the most part the same type of cost function is used for neural networks. We usually use the principle of maximum likelihood for most cases, such as when our parametric model uses a distribution p(y | x;θ), where the relationship between the input (training data) and output (model's predictions) of the model as the cost function. A sometimes simpler approach used, where we are predicting some statistic of y conditioned on x, instead of predicting a complete probability distribution over y.