Time

According to Wikipedia, Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. We can further read that there are two contrasting viewpoints on time.
One view is that time is part of the fundamental structure of the universe – a dimension independent of events, in which events occur in sequence. Isaac Newton subscribed to this realist view, and hence it is sometimes referred to as Newtonian time. The opposing view is that time does not refer to any kind of "container" that events and objects "move through", nor to any entity that "flows", but that it is instead part of a fundamental intellectual structure (together with space and number) within which humans sequence and compare events. This second view, in the tradition of Gottfried Leibniz and Immanuel Kant, holds that time is neither an event nor a thing, and thus is not itself measurable nor can it be travelled.
Aristotle believed that time correlated to movement. In his opinion, time do not exist on its own but is relative to motion of objects. He also believed that time is related to the motion of celestial bodies which is why it seems like something non-existent.

The problem with people today is that they are willing to question everything without much of a thought. Just because something is not obvious it does not make it a lie. Life is full of uncertainty.
In the book A brief history of time, Hawking describes Aristotle and Newton's belief in absolute time, meaning that time can be measured accurately regardless of the state of motion of the observer. However, Hawking writes that this commonsense notion does not work at or near the speed of light. He mentions Danish scientist Ole Rømer's discovery that light travels at a very high but finite speed through his observations of Jupiter test showed light traveling in waves moving at a fixed speed. Assuming that there is no absolute time, light travels with a finite speed no matter what the speed of the observer is.
Time in physics is operationally defined as "what a clock reads". In simple words, the repetition of events that appear cyclicaly. In earlier days, before the atomic clock were invented, people were measuring time relying on periodic events like the apparent motion of the sun across the sky, the phases of the moon, and the passage of a free-swinging pendulum. 
I have heard on social media about the conspiracy theory that says about the 13th month gone missing. There is some truth in that. The 13th month is indeed gone. The reason for it though is slightly different than some might expect. We can read on Wikipedia that Artifacts from the Paleolithic suggest that the moon was used to reckon time as early as 6,000 years ago. Lunar calendars were among the first to appear, with years of either 12 or 13 lunar months (either 354 or 384 days). Without intercalation to add days or months to some years, seasons quickly drift in a calendar based solely on twelve lunar months. Lunisolar calendars have a thirteenth month added to some years to make up for the difference between a full year (now known to be about 365.24 days) and a year of just twelve lunar months. 

Why do we feel like some days time is speeding and other days is moving extremely slowly?
That is rooted deep within us. Let's imagine that we represent movement and every time we have our pattern changed we get broken a little. As we go through life we live by pattern whether we want it or not. Yes, even the most spontaneous person live by pattern (only a bit more complicated one). On our way we meet other people that have different pattern to ours. The moment of the meeting represents connection of two patterns which creates a junction. Now, we have two patterns that has been broken. At that moment, the time either goes slower or faster. When it's something that we are used to, time passes slower while when it's something that is far from we know, time goes faster. Add to that equation our likes and dislikes and you can now see how it works. Everything within us runs our perception making us wonder about deeper meaning of the world around us while in the end it's all us.

The concept of time is more simple that we might want it to be. We are so used to making our life harder and complicated that we forgot that it doesn't have to be this way with everything.