The Transformer is a neural network architecture that processes a whole sequence at once using self-attention instead of stepping through it one token at a time. It solves the problem that recurrent networks could not parallelise training and struggled to relate distant positions in a sequence. It exists because attention alone turned out to be sufficient for translation quality, and because removing recurrence made training scale across many accelerators. Almost every large language model in use today is a Transformer variant, which makes it the single most load-bearing node in this graph.
An LSTM is a recurrent network whose internal gates decide what to keep, what to write and what to output, which slows the decay of information across many steps. It solves the vanishing-gradient problem that made plain recurrent networks forget the beginning of a long sequence. It exists because much of the useful context in language is far from the token being predicted. It was the standard sequence model before attention replaced it.