Neural Networks for time series
The type of neural networks you are looking for to predicting timeseries are called recurrent networks; the previous neuron states in the network affect the new neuron states. Examples of such recurrent networks are LSTM, GRU and NARX.

If you have the data from t0 to t10, and you want to predict t11, then you need to input the values from t0- t10 one by one into the network. You have to train the network so that the output of tn is tn+1. So after you have inputted t10, the output will be the values predicted for t11.

Real life example: I want a network to keep on decreasing it's output until it has reached 1, and then make it output 0 to start over again:

in: 0.0, out: 0.2
in: 0.2, out: 0.4
in: 0.4, out: 0.6
in: 0.6, out: 0.8
in: 0.8, out: 1.0
in: 1.0, out: 0.0


And that is what your training data should look like

If it is still to complicated, I recommend you to play around with this front-end neural network library for the browser: Neataptic. It is very easy to fiddle around with. Example