Today in chemistry class we were discussing Organic Chemistry. We discussed what organic compounds basically are and then I asked the teacher whether $\ce{CO_2}$ is organic or not. She told that it is as it contains carbon and oxygen with a covalent bond. I told her it can't be as it is not found in animals (naturally). I am very confused about it.
I need some good reasons to agree with either explanation. (I have searched the internet already but found no great insights as of now).
It is entirely arbitrary whether you call it an organic compound or not, though most would not.
The distinction you make that organic compounds should be found in living things is not a useful criterion. Moreover you are wrong that carbon dioxide isn't: it is both made and used by living things. Animals make it when they metabolise sugars to release energy; plants consume it when they build more complex organic molecules through photosynthesis. In fact most organic molecules are, ultimately, derived from $\ce{CO2}$.
Even more importantly most molecules considered organic are neither made by nor are found in living things. Chemists make new carbon compounds all the time (tens of millions in the history of chemistry) and most have never been made by animals or plants.
The organic/inorganic terminology is mostly very simple: covalent compounds containing carbon are organic. The only fuzzy area is around very simple molecules like $\ce{CO2}$ where the distinction doesn't matter much. So we would not normally think of diamond or silicon carbide as organic. But we might (though many would not) call calcium carbide organic because it contains a $\ce{C2}$ unit with a carbon-carbon triple bond.
However since the terminology is mostly very obvious and also somewhat arbitrary, it isn't worth much argument to sort out those very simple but awkward edge cases.
There is no exact definition of "organic" compound, although you can say that organic compounds must contain carbon.
There is no requirement that organic compounds can be found in living things, although the name organic comes from the fact that the first compounds in this class that were discovered did come from living things. That said, $\ce{CO2}$ is in living things - all metabolic processes either consume or produce $\ce{CO2}$ (with the exception of some thermophiles).
In general, ionic and "simple" molecular compounds that contain carbon are not called "organic," partly for historical reasons, but also partly because the category "organic" is more than just a name - it describes an entire set of bond types, reaction patterns, naming rules, and structure types. This is why there is no exact, simple definition - it is hard to draw a boundary around all of that and say "these other things don't belong."
Taking your example, $\ce{CO2}$ is not normally called "organic," even though it contains carbon, it is produced and consumed in biochemical reactions, it has similar properties to other organic molecules, and organic bonding principles can describe it. The reason for this is probably because historically, $\ce{CO2}$ was discovered and named before organic chemistry existed as a field.
Some people say that in order to make it easy, any carbon-containing covalent compound is "organic" (your teacher is probably in this group). Some people say "any compound containing carbon and hydrogen" should be the rule. Both of these rules have exceptions, though, so they can't be used as exact definitions.
Carbon dioxide is indeed strictly speaking organic, as it is a compound containing carbon. Your claim that it doesn't naturally occur in living organisms is not correct. For instance, it is the product of breathing.
That said, some chemists do not consider it to be organic as it has no hydrogen in it- a critical component of just about every organic compound.
You have to remember that the term "organic" is just a vague definition given by humans to some types of chemicals. Nature does not always play by our rules, especially when you're talking about borderline cases such as carbon dioxide.
$\ce{CO2}$ is commonly considered inorganic. You will rarely find a textbook on organic chemistry that discusses $\ce{CO2}$ and its properties. On the other hand every textbook on basic inorganic chemistry does.
Interestingly, urea $\ce{(NH2)2CO}$ historically was considered organic and its synthesis from inorganic reactants (ammonium cyanate, F. Wöhler, 1828) is widely regarded as the beginning of the end of vitalism.
