I know this might be a stupid one, but could anyone explain the whole process for me,

X^2= 7 (mod 3^3)

2. $\displaystyle x^2 \equiv 7 \ (\text{mod } 27)$

$\displaystyle \Leftrightarrow x^2 \equiv 169 \ (\text{mod } 27)$ (adding the modulus to 7: 7 + 6(27) = 169)

Why 169? 169 is a perfect square

Does that help now?