On a macro scale, neuroscientists now know (more or less) where memories are 'stored' in the human brain. The hippocampus, the amygdala, the striatum and the mammillary bodies (for example) are known to be involved in some way, because individuals who suffer damage (either by injury or disease) to those areas are prone to memory loss of various kinds.
Different modes of memory (short-term, long-term, motor memory etc) have been extensively classified and described, but researchers aren't yet in complete agreement about the models.
On the small, neuronal scale - at 'component level' so-to-speak – the mechanism(s) by which complex memories are stored and retrieved are still a complete mystery.
Further reading: Wikipedia