What is a square root?
As I don't fully understand your question, I will divide my answer into three parts: the definition of a square root, how to understand it, and algorithms to calculate it.
DEFINITION OF A SQUARE ROOT
To cite Wikipedia,
In mathematics, a square root of a number $a$ is a number $y$ such that $y^2=a$, in other words a number $y$ whose square $(y \times y)$ is $a$. For example, $-4$ and $4$ are both square roots of $16$ because $(4)^2=(-4)^2=16$.
Every non negative real number $a$ has a unique non-negative square root, called the principal square root, which is denoted by $\sqrt{a}$ where the $\sqrt{\square}$ sign is called a radical sign or a radix. For example, the principal square root of $9$ is $3$, denoted $\sqrt{9}=3$, because $3^2=9$ and $3$ is non negative. The term whose root is being considered is known as the radicand. The radicand is the number or expression underneath the radical sign, in this example $9$.
HOW TO UNDERSTAND IT
A way to approach this is thinking by reversal. What you do, someone can undo.
A $\color {blue}{\text{summation}}$ is the reverse of $\color {blue}{\text{subtraction}}$. Indeed, $\color {blue}{\text{subtraction}}$ can 'undo' a $\color {blue}{\text{summation}}$, and $\color {blue}{\text{summation}}$ 'undoes' a $\color {blue}{\text{subtraction}}$.
$\color{green}{\text{Division}}$ is the reverse of $\color{green}{\text{multiplication}}$. $\color{green}{\text{Division}}$ undoes a $\color{green}{\text{multiplication}}$, $\color{green}{\text{Multiplication}}$ can be 'undone' with a $\color{green}{\text{division}}$.
Think of a $\color {orange}{\text{square root}}$ as a reversal of a $\color {orange}{\text{square}}$: you can "undo" a $\color {orange}{\text{square}}$ with a $\color {orange}{\text{square root}}$, and a $\color {orange}{\text{square}}$ can "undo" a $\color {orange}{\text{square root}}$. This may lead to some confusion, as $x^2$ is not invertible. To clarify, think of the square root of $16=4^2=(-4)^2$. A square root is not a function. Thus taking the square root of $16$ undoes the square, yielding two values- it becomes $4$, $-4$.
So to summarize $$\color {blue}{\text{summation}} \leftrightarrow \color {blue}{\text{subtraction}}$$$$\color{green}{\text{Division}} \leftrightarrow \color{green}{\text{multiplication}}$$ $$ \color {orange}{\text{square root}} \leftrightarrow \color {orange}{\text{square}}$$
ALGORITHIMS TO FIND IT
If you want to determine if it is whole or irrational, if a integer is between two squares, then it is decimal. In your example, since $351^2=123201<123456<123904=352^2$, it is irrational.
Of course, there are more clever arguments. For example, if $3$ divides $n$ but $n$ is not divisible by $9$, this implies $n$ is not a square. This is true in case of $n=123456$. Other examples include $$n \equiv 2 \pmod 3 \Rightarrow n \quad \text{is not a square}$$ $$n \equiv 2,3 \pmod 5 \Rightarrow n \quad \text{is not a square}$$ But note these methods are not always easy-and sometimes require luck.
A operation for finding (or approximating) square roots exists, not to different from long division. A simple one can be seen here.
Of course, the quickest way to approximate it is using the sequence {$x_{n}$} which is defined as follows.
$$x_{n+1}=\frac{1}{2}(x_{n}+\frac{c}{x_{n}})$$
It is not to hard to see that $\lim_{x \rightarrow \infty}x_{n}=\sqrt{c}$. This sequence actually converges pretty quickly, so this is useful, which is equivalent to Newton's Method in $x^2-N$.
There are, of course, other algorithims to calculate the square root. And there are some rather exclusive ones as well. For example, $$\lim_{n \rightarrow \infty}{\frac{F_{n+1}}{F_{n}}+\frac{F_{n}}{F_{n+1}}}=\sqrt{5}$$ Where $F_{n}$ is the Fibonacci Sequence. But not converge quite as quickly as the one above.
More discussion on the definition, and the algorithims can be found in the links provided.
I think you're confusing the definition of a square root with the algorithm needed to compute it. In mathematics, there are many concepts that we can define, but where it may be hard to actually find some object satisfying the property.
A square root of a number $n$ is defined as any number $x$ that satifies this property: $$x^2 = n.$$
For example, we can verify that $2$ and $-2$ are square roots of $4$.
Often we talk about the square root, which is an abbreviation for the term principal square root. By that we mean the simple convention that $\sqrt{4} = 2$ (and not $-2$). But that is just a convention.
Actually finding a square root is more complex. Many algorithms will use somewhat higher mathematics. But the gist of it is: take a number $a$ whose square is less than $n$ and another number $b$ whose square is bigger than $n$ and then you know that the square root of $n$ must be between those two. Repeat until you have enough accuracy as needed (you will rarely ever be done, since most square roots are irrational).
If you think at the square power $l^2$ as the area of the square of side $l$, than the square root is to find the side of a square of given area. This seams the more intuitive interpretation of the square root for positive real numbers.