A prime number is a natural number greater than $1$ that has no positive factor other than $1$ and itself.

A composite number is a natural number greater than $1$ that has at least one factor other than 1 and itself, in other words, is not prime.

Consider a grid of consecutive integers ranging from $2$ to $n$; we exclude $1$ from the grid because it is neither a prime nor composite number.

Let $p$ be the first prime number, $2$.

Mark off its multiples ($p\left(p\right)\, p\left(p\+1\right)\, p\left(p\+2\right)....$) ranging from $p^2$ to $n$ as composite numbers (mark $p$ itself as prime).

Take $p$ to be the next unmarked number on the grid, mark it as prime and mark all of its multiples ranging from $p^2$ to $n$ as composite numbers.**

Repeat step 4 until there are no more unmarked numbers in the grid less than or equal to $\sqrt{n}$.  For example, while  $p \le \sqrt{n}$.

Mark the remaining unmarked numbers in the grid greater than $\sqrt{n}$ as prime numbers.