Is this the correct cayley table for the units of integers mod 21?
how would you find the order of each element?
What ye did looks correct.
To find the order of, say $\displaystyle [5]_{21}$ you need to take exponents until you reach $\displaystyle [1]_{21}$.
$\displaystyle [5]_{21}^2 = [4]_{21}$, $\displaystyle [5]_{21}^3 = [20]_{21}$, $\displaystyle [5]_{21}^4 = [16]_{21}$, $\displaystyle [5]_{21}^5 = [17]_{21}$, $\displaystyle [5]_{21}^6 = [1]_{21}$.
Therefore the order of $\displaystyle [5]_{21}$ is $\displaystyle 6$.