A false awakening is an event in which someone dreams they have awoken from sleep. This illusion of having awakened is very convincing to the person. After a false awakening, people will often dream of performing daily morning rituals, believing they have truly awakened.
A dream in which a false awakening takes place is sometimes colloquially referred to as a "double dream", or a "dream within a dream". A false awakening may occur either following an ordinary dream or following a lucid dream (one in which the dreamer has been aware of dreaming). Particularly if the false awakening follows a lucid dream, the false awakening may turn into a ‘pre-lucid dream', that is, one in which the dreamer may start to wonder if they are really awake and may or may not come to the correct conclusion.
A false awakening has significance to the simulation hypothesis which states that what we perceive as "true" reality is in truth an illusion as evidenced by our minds' inability to distinguish between reality and dreams. Therefore, advocates of the simulation hypothesis argue that the probability of our "true" reality being a simulated reality is affected by the prevalence of false awakenings.
I have had these. You also see this as a gimick in movies a lot, mostly horror movies. I think Carrie had a segment at the end like this but perhaps memory fails.