Recurrent Neural Network (RNN) is one of the Neural Network ( Deep Learning ).
RNN deals with the data with the order. For example, writings have orders of letters.
The output of a layer is used as input of next layer.
LSTM is Long Short Term Memory.
LSTM is one of the RNN. It can use the output not only just before the layer. And it can use complicated pattern.
The idea that uses the output of old layer also used AR model and State Space Model .
RNN uses this idea for non-linear model.
The relatiionship between Multi-Regression Analysis and AR model is similar to the relationship between Neural Network and RNN.