No one has figured all this out yet. However, we know that all mental processes are the result of the actions of the neurons because of things like the correlation of neural activity with mental states as picked up on an EEG(alpha waves, beta waves, etc.), and because of the loss of functions when neurons are damaged. After a stroke, for instance, in which the neurons governing motor control on the right side of a persons brain are starved of oxygen by blockage of a blood vessel, that person can no longer move a muscle on their left side. Likewise, in the case of severe hippocampal damage a person can no longer form long term memories. In addition there is a process by which a person's brain can be completely stopped by lowering the temperature of their blood beneath a certain point where neurons cease to fire altogether and all mental and autonomic functions likewise cease. This can be maintained for something like 90 minutes, the person's blood then warmed to normal temperature, and all previous functions return.
Exactly how the actions of neurons add up to any particular function is not precisely known, but we are certain that all functions are dependent on the actions of neurons. If I disturb the actions of a particular part of your brain called the thalamus, for example, you will instantly lose consciousness. Were your thalamus to suffer outright severe damage you would go into a coma with no hope of ever regaining consciousness.
I think it's safe to say that a particular memory depends on what neurons fire, in what sequence, and possibly in what rhythm. I don't think anyone can explain how those neurons firing in that sequence at that rhythm add up to the experience of memory, but as research progresses I think we'll get closer and closer to figuring out what is going on.