Matlab ntstool NAR timeseries response

Good afternoon, I am using the Matlab (GUI) 'NTSTOOL' for a NAR-based neural network. I have one input variable (time-series) and I would like to predict future values of my time-series given 'n' number of previous samples. I have successfully loaded the data via the GUI and partitioned training, validation and test data. When the NN finishes training/validation/testing, I can select various plots, e.g. 'time-series response'. However, it is not clear to me what the 'time-series response' plot shows. I can see it plots 'targets' (which are my input data) and 'outputs' (the output from the neural network); but I do not understand the mechanics of how my input data is being processed by the neural network in order to produce these results shown in this graph. During training, is the input data (target data) processed element by element, or does it get processed in parallel with respect to the number of 'delays'? It also isn't clear to me what the 'performance' plot shows. I understand how error (mse) would converge towards zero during the 'training' phase (and potentially even during 'validation') but why so during 'testing'? I've read a lot(!) of the Matlab help pages, and cant seem to get this answer. Any help greatly(!) appreciated...