Let's see how inertial pseudoforces (like the centrifugal pseudoforce) arise in the theory of Newtonian mechanics.1
Rule: Newton's laws pre-suppose that you are working in an inertial frame.
The first rule can be regarded as a way of defining or identifying those frames (assuming you can identify forces, anyway).
Those three laws don't give any direct advice about doing physics in non-inertial frames.
Way to identify real forces
If you examine a single physical situation from several frame of reference,2 some "forces" that you see may change their direction or magnitude between frames, while others will remain stubbornnly the same.3 The ones that are always the same are are "real".
Observation: Sometimes it's nice to do physics in non-inertial frames.
If you are sitting in a vechicle that is moving and put a nice cup of coffee onto a tray. It sits there, at rest relative you. In class we'd use that kind of observation ("It's just sitting there.") to identify things that are in equilibrium, and then we would assert (by way of the second law) that the sum of the forces acting on it is zero.
And if your vehicle is in uniform motion that identification would be correct. but if your vehichle is accelerating (changing speed, going arouind a curve, both...) it's formmaly incorrect. The cup is also accelerating.
But we might want to go ahead with our usual analysis anyway. That's where intertial pseudofoces come in.
Plan: let's lash it up!
Our lash-up scheme is very simple. We start with the physics given to us by Newton's laws; move any inconvenient accelerations from the RHS to the LHS; and call the new terms on the LHS "forces".
That's the whole shebang.
Round-about example
Consider for concretness a car driving $20\,\mathrm{m/s}$ around a curve with radius $10\,\mathrm{m}$. We suppose that the thust provided by the drive train balances the drag and rolling friction and so that the only unballanced force is static friction directed toward the inside fo the circle.
Set-up Newton's laws in an inertial frame
\begin{align}
\sum_i \vec{F}_i &= m \vec{a} \\
\vec{F}_\text{thrust} + \vec{F}_\text{drag} + \vec{f} &= m \frac{v^2}{r}\left(-\hat{r}\right) \\
\vec{f} &= -m \frac{v^2}{r} \hat{r} \;.
\end{align}
Decide we want things "at rest" in our non-inertial frame to be "in equilibrium", so move that inconveinent term to the other side.4
\begin{align}
\vec{f} + m\frac{v^2}{r}\hat{r} &= 0 \;.
\end{align}
This is apurely formal, mathematical operation.
Give the term we just moved a name $\vec{F}_\text{centrifugal} = m \frac{v^2}{r}\hat{r}$, so that the equation now has two "forces" it in
\begin{align}
\vec{f} + \vec{F}_\text{centrifugal} &= 0 \;.
\end{align}
Now, because of the subtraction process this newly created "fake" force has the opposite direction tha the real acceleration had.
Tradiational, however we write this force in terms of the rotational velocity $\Omega$ of the frame $\vec{F}_\text{centrifugal} = m r \Omega^2 \hat{r}$.
More generally
There is a standard, fully generaly way to deal with complicated non-inertial motion. It develops four pseudoforces, each related to a specific kind of behavior.
A pseudoforce related to observers with straight-line acceleration $\vec{A}$ with respect to an inertial frame (oddly this one has no tradiational name; I sometimes call it the "seatbelt pseudoforce"):
$$ \vec{F}_\text{seatbelt} = -m\vec{A} \;.$$
The "centrifugal" or "centrifical" pseudoforce related to observer in rotating with angular velocity $\Omega$ with respect to an inertial frame. This affects all objects including those at rest with respect to the observer. We just computed it5
$$ \vec{F}_\text{centrifugal} = -m \vec{\Omega} \times (\vec{\Omega} \times \vec{v}_b) \;. $$
The "Coriolis" pseudoforce which is also related to rotations but is observed only for objects that move wth velocity $\vec{v}_b$ in the non-inertial frame.
$$ \vec{F}_\text{Coriolis} = -2m \vec{\Omega} \times \vec{v}_b \;, $$
where $\vec{v}_b$ is the velocity observed in the non-inertial frame.
The "Euler" pseudoforce which arises for observers experiencing angular accelration relative an inertial frame.
$$ \vec{F}_\text{Euler} = -m \frac{\mathrm{d} \vec{\Omega}}{\mathrm{d}t} \times \vec{x}_b \;.$$
You can find detailed mathematical developments of this stuff in typical upper-division or graduate mechanics textbooks.
But ... do pseudoforces exist, already?!?
Yes? no? Depends?
This is, fundamentally, a philosophical question that turns on how you understand "existing". Their magnitude and direction depend on the frame from which you view a physical interaction, which definitely sets them apart from those "real" forces that don't have that property.
For myself, I make a point of maintaining a strong distinction between "real" and "pseudo" forces. But I am perfectly happy to work in non-inertial frames when that makes my life easier.
Bonus: What are people going on about gravity for?
Did you notice that all the pseudoforce definitions I gave have exactly one factor of the object's mass in them? That means that all objecs experince the same "centrifugal" or "Coriolis" accelration, which is supiciously like the rule that everything falls with the same acceleration.
As it turns out, when Einstien found with way through the weeds to general relativity he found tha he had created a theory in which gravity was also an inertial pseudoforce (and it such as heck is "real" enough for day to day purposes, isn't it?).
In general relativity when you are standing at rest in the lab you are observing the world from a non-inertial frame. The inertial frame is what you would see standing on (or ratehr, floating next to) the ball the instructor just dropped.
1 In this discussion I'm going to consistently identify that class of apparent forces tha arise in non-inertial frams as "pseudoforces" simply to set them apart from those "real" forces that appear in all frame inertial or otherwise.
2 It is important that I emphasize: one set of physical events as seen by observers with different states of motion. Not multiple events characterized by different motion of the participants.
3 Here the direction and magnitude are to be identified as their intrinsic values. Don't worry abrout changing components, just about changing nature.
4 The 'at reast" and "equilibrium" that appear here are purely for motivation. You shouldn't read any implication that this analysis only applies to things at rest in the non-inertial frame. Indeed the Coriolis forces is only interesting for this in motion in the non-inertial frame.
5 So far, I've written the centrifugal force in it's simplest form (figuring the directions has been easy because they are just "in" or "out"). For consistency with what follows I'm using
$$ \vec{F}_\text{centrifual} = -m \vec{\Omega} \times (\vec{\Omega} \times \vec{x}_b) $$
with $\vec{\Omega}$ being the vector angular velocity and $\vec{x}_b$ begin the position of the object in a coordinate system with it's origin at the axis of rotation. Trust me. It's the same thing.
