The keyword is "cycle structure". (I will leave it as an exercise for you to discover the meaning of "cycle structure of a permutation".)
Exercise 1: If $n$ is a positive integer and if $\alpha,\beta\in S_n$, then $\alpha$ and $\beta$ are conjugate in $S_n$ if and only if $\alpha$ and $\beta$ have the same cycle structure.
There are three different possible cycle structure for elements of $S_3$ (if one includes the cycle structure of the identity element of $S_3$). The two elements in $S_3$ of order $3$ share one cycle structure, the three elements in $S_3$ of order $2$ share a different cycle structure, and finally the identity element in $S_3$ has the trivial cycle structure.
Exercise 2: Prove that a subgroup of $S_3$ of order $2$ is never a normal subgroup of $S_3$. How many subgroups of $S_3$ have order $2$? (Hint: Exercise 1 is relevant.)
Exercise 3: Prove that there is exactly one subgroup of $S_3$ of order $3$ and that this subgroup of $S_3$ is a normal subgroup of $S_3$.
Exercise 4: Prove that in addition to the trivial subgroup and the entire group $S_3$, you have determined all subgroups of $S_3$ in Exercise 2 and Exercise 3. (Hint: Lagrange's theorem.)
Challenge Exercise: If $n$ is a positive integer factor of $24$, then determine the number of normal subgroups of $S_4$ of order $n$. (Hint: a basic knowledge of group actions is very helpful and a knowledge of Sylow theory probably trivializes the question.)
I hope this helps!