Layers of Lucidity
"Layers of Lucidity" is a concept invented by Reece Jones to describe the level of lucidity experienced in a dream. I don't particularly like the term "layer" but I think his ideas make sense, and this is the best method I've seen for classifying lucid dreams.
For a detailed explanation of the layers you should probably go and watch his video on the layers of lucidity on YouTube.
Non-Lucid
If a layer is not recorded against a dream, it means that it is a regular dream and not in any way lucid.
Layer 0
This is when you dream about lucid dreaming, but don't become lucid. In your dream you are aware that lucidity exists, but don't recognise that you are dreaming.
Some examples of this would be reading a book about lucid dreaming or speaking to a dream character about lucid dreaming, but not becoming lucid.
Layer 1
This is when you realise that you are dreaming, but it doesn't last. Most commonly this is because you wake up straight away, but it can also be because you have a false awakening, or because your lucidity fades (normally with the dream temporarily fading out, and then back in, but this time with you being unaware that you are dreaming).
The key thing here is that the lucidity is very short lived.
Layer 2
In layer 2 you realise that you are dreaming, but fail to understand the implications of this. You don't really recognise that you are still in bed, and that everything is being generated in your mind, you have no control and will still tend to use dream logic when making decisions.
Layer 3
Layer 3 is the layer where you start to really achieve lucidity. You have your waking conscience in the dream world, and comprehend that the whole thing is a simulation created inside your head. You still don't have full control though: you have to obey the rules of physics inside the dream.
Layer 4
This is the final layer of lucidity, where you have complete control over everything. You can manifest objects, use supernatural powers and generally bend the dream world to your will.