Why is the square root of a negative number impossible? What is the algorithm to get the square root of a number?
Why does it not accept / allow for negative numbers?
I've always been told that the square root of minus one does not exist by my maths teacher, but I don't really get why?

First, we have to specify what set of numbers you are working in, because otherwise, the question is meaningless.


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*Does the reciprocal of $2$ exist? If you're working in $\mathbb{N}$, then no, it does not. But if you're working in $\mathbb{Z}$, it does.

*Does $x$ such that $x \cdot 2 = 1$ exist? There is no element of $\mathbb{Z}$ that satisfies this, but there is one in $\mathbb{Q}$.

*Does $\sqrt{2}$ exist? In $\mathbb{Q}$, no, but in $\mathbb{R}$, yes.


So it is very important to specify which set of numbers you are talking about!

So let's change your question to: "does there exist an $x \in \mathbb{R}$ such that $x^2 = -1$?" Since we've now specified what set of numbers we're talking about, this has an actual answer. No.
We know that the real numbers have an order $<$ with these properties:


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*For all $a,b \in \mathbb{R}$, exactly one of these is true: $a < b$, $a = b$ or $a > b$.

*If $a > 0$ and $b < c$, then $ab < ac$.

*If $a < 0$ and $b < c$, then $ab > ac$.


Pick any $x \in \mathbb{R}$. We use property $1$ to split this into cases:


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*$x > 0$: By property $2$, we know $x \cdot x > x \cdot 0$, and so $x^2 > 0$.

*$x = 0$: Computation shows $x^2 = 0$.

*$x < 0$: By property $3$, we know $x \cdot x > x \cdot 0$, and so $x^2 > 0$.


In all cases, $x^2 \ge 0$. And since $-1 \not\ge 0$, we know that no square root exists in $\mathbb{R}$.

But is there a system of numbers in which there is a square root of $-1$? Yes; they are called the "complex numbers".
The complex numbers can be defined in lots of ways, but for our purposes, they are numbers of the form $a + bi$, where $a$ and $b$ are real, and $i$ is a number where $i^2 = -1$. Note that $i$ is not a real number! But because we're making a new number system, we can make up whatever we want.
By construction, $i = \sqrt{-1}$. Note that all real numbers have squares now: let $x = a + 0i$. If $a \ge 0$, we just take the square root as normal: $(\sqrt{a})^2 = a$. But if it is negative, we consider $\sqrt{-a} \cdot i$. When you square this, you get $a$.

A side note: like the real numbers, there are two square roots for each number. For example, $i^2 = -1$, but so does $(-i)^2$. With the real numbers, we can choose the positive one as our "principal" square root. But the complex numbers don't have $<$ anymore, so the expression $\sqrt{-1}$ is ambiguous. To avoid this, we usually don't use $\sqrt{x}$ when working with complex numbers, unless we know for certain that $x$ is real and non-negative. Our definition of $i$ changes to: $i$ is a number such that $i^2 = -1$. 
One might ask: how do we tell which square root is $i$ and which is $-i$? Well... we don't. If you swapped $i$ and $-i$ everywhere in the world, then true statements would still be true, and false statements would still be false. So our "choice" doesn't really matter, and we just say "$i$ is one of the square roots, and $-i$ is the other". If you continue in mathematics, you will recognize $\mathbb{C}$ as a field extension, and you can formalize what is meant by "it doesn't matter which is which".
