But cells can affect gene expression throught multiple mechanisms, so that part of the DNA gets activated or de-activated.

You can use those mechanisms to react to the environment in various ways. For instance when you see chemical signals A and B, start making protein C. Or, when you see some signal, make the neuron more sensitive at the synapse (add receptors).

You can see it as storing state in a sort of memory, but that's pushing the computer analogy to its limits. Cells are pretty unlike chips :)