The child is 2 years old.
When do children start understanding the concept of today, yesterday, and tomorrow? When do they start understanding that the days are passing?
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If the answer is vague, that might be because the question is also vague, perhaps not purposely, but it is, because there is a lot that relates to different kinds of time and the theory of the mind.
For a long time, and the vast majority of people still believe this, people believed that of all animals, only humans had a concept of time, that animals live in a permanent present, or now (the fact that some animals can keep a beat essentially disproves this.)
[The perception of time] is not an isolated module, but depends on the sophistication of other cognitive capacities, including self-awareness, meta-representation, mental attribution, understanding the perception- knowledge relationship, and dissociation of imagined mental states from one's present mental state. These capacities are also important aspects of so-called "theory of mind", and they appear to mature in children at around age four.
However, it can be argued that whenever a child can understand the concept of waiting their turn in a game, clapping to the rhythm of music (which depends on the accurate interpretation of past intervals and future intervals), counting to 10 in hide-and-seek or even when the concept of object permanence develops, there exists a distinct understanding of concept of time. It isn't necessarily exactly like ours - actually the perception of time continues to change throughout life - but it's there.
The concept of "now" vs. "later/before" occurs in the pre-verbal stage. The concepts of "today and tomorrow" appear much earlier than the adult understanding of time, which as stated above occurs around the ages of 4 or 5.
The truth is that certain things do follow certain timetables, and other things do not. In the usual model of childhood development, probably the most influential person in the past century is Jean Piaget (1896-1980) was a developmental psychologist/philosopher known for his theory of cognitive development. The problem is that many children don't follow the 'rules', that is, there's more variability to development than the linear development model allows. Kids can have a moment of sophistication followed by a moment of something 2 stages back, i.e., a 1:1 relationship between stages and ages isn't always supported by reality.
When your child is told that they will go to the DisneyWorld tomorrow, and they wake up the next day asking "when are we leaving?", then trust that they understand.
For my kids, that happened well before age four.
Mental time travel and the evolution of the human mind
The Time of Our Lives: Life Span Development of Timing and Event Tracking
Emerging Minds:The Process of Change in Children's Thinking, Oxford University Press
For the record I am looking at this question as two different sub question. First question is when do humans gain temporal awareness? and when are they are they able to make sense of this temporal awareness? (I.E when are humans able to construct a meaningful relationship between their temporal awareness and a language). Also, I think I should make this abundantly clear how I define time and awareness may be abundantly different than you so when you read the next paragraph please keep this in mind.
So first things first when do we as humans gain temporal awareness? The earliest suggestion I can find is about 1 month. However, I don't think that particular study is strong evidence. In essence that study demonstrated that infants are able to "learn" a given temporal interval between two events, but considering it was behavioral study it's hard ensure what was observed was indeed awareness.
Moving on, a study done with 10 month year infants has shown an infants brain reacts in the same way as adult's to temporal deviations. But, the fun doesn't stop there as a follow up study by the same authors discovered in essence that infants automatically detect temporal irregularities when given repetitive stimuli. So what do these studies mean? It means given a repetitive stimuli like say the beep of a metronome the infant in question will be able perceive any beeps that are not in line with the metronome as irregular, which gives the suggestion that the infant is able to "keep time" in to determine that the irregular beep is "out of time"... (or at least that is how I understood the results of the study) If you want a really good review (which my answer is partially based on) of the evidence I suggest you buy/read this 2012 review I have linked in.
However, it is probably not until 2-3 years old when children are first able to... lets say communicate that they have temporal awareness. Whether they understand is dependent on the child. For example, in this study which verbally tested children between the age of 3-9 on their understanding of time through story telling it was discovered that it usually not until 8 or 9 years of age do children consistently apply time related words correctly. Which suggests as the author noted that perhaps up until 8 or 9 children are experimenting and testing their new found linguistic knowledge. Also, you have take in consideration how schools and parenting play a role into this, but that kinda goes out the scope of this question.