Memory is often associated with Recurrent Neural Network (RNN), but that is not entirely an accurate association. An RNN is really only useful for storing a sequence of events or what you may refer to as a temporal sense, a sense of time if you will. RNNs do this by persisting state back onto itself in a recursive or recurrent loop. An example of how this looks is shown here:

What the diagram shows is the internal representation of a recurrent neuron that is set to track a number of time steps or iterations where x represents the input at a time step and h denotes the state. The network weights of W, U, and V remain the same for all time steps and are trained using a technique called Backpropagation Through Time (BPTT). We won't go into the math of BPTT and leave that up the reader to discover on their own, but just realize that the network weights in a recurrent network use a cost gradient method to optimize them. 

A recurrent network allows a neural network to identify sequences of elements and predict what elements typically come next. This has huge applications in predicting text, stocks, and of course games. Pretty much any activity that can benefit from some grasp of time or sequence of events will benefit from using RNN, except standard RNN, the type shown previously, which fails to predict longer sequences due to a problem with gradients. We will get further into this problem and the solution in the next section.