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.
