Conscious memory is the memory in your mind which you have conscious access to. It is the memory that keeps all the “important” stuff like your bills, your car payments, your appointments etc. All our recent and important memories and emotional aspects are kept in your conscious memory. This could be a time frame of 1 month, 1 year, 5, 10 or more years and it's through these thoughts that we sift and rummage through on a daily basis.
In order for something to register in your conscious memory, it has to be first experienced and labelled as “important” in some way by your mind.
Our subconscious memory keeps all the stuff that isn't “important” but you did experience in some way. You could look at this as all the stuff that gets shuffled aside by your mind and thrown in the garbage pile.
For instance, if I were to ask you how dirty your socks were when you took them off 5 days ago, chances are you wouldn't be able to remember this.
Believe it or not, your mind did register this. It just put it in the “subconscious memory” pile. If you enter a relaxed state, and become more in-tune with your subconscious mind, sometimes thoughts and memories from this area will surface and come to your awareness. This is because thoughts are connected in a chain like structure, where one thought leads to the next. Have you ever had the experience of trying to remember something all day, but not being able to? Then, all of a sudden having that thing pop into your mind later on when you weren't expecting it?
As I said before, everyone dreams every night, we simply don't remember our dreams. You see, when we're dreaming, our conscious mind is completely turned off. Our dreams are a production of our subconscious mind, so we're only aware of them when we're having them, and the memory of these dreams DOES NOT enter our conscious memory while we're dreaming. Dreams automatically enter our subconscious memory as we're having them. Therefore, if you try to remember your dreams by accessing your conscious memory, you're out of luck!
I learned to term the memories of these dreams: shadow memory. It is memory that is not linked to our conscious mind in anyway, the source of these memories is purely the subconscious mind, therefore, the chance that these memories will suddenly pop up in your waking life are next to none, and most of the time you will never become consciously aware of these memories for the rest of your life.