Given the following...
Let $Q=\{1,2,3,4,5,6,7,\dotsc,26\}$ and let $P$ be the non-empty subsets of $Q$. For $A\in P$ and $B\in P$, $A\sim B$ iff there is a bijection $f\colon A\to B$.
How would I describe (not list) the set $A\in P$ when $A\sim\{1,2,3,4,5\}$?
And how many equivalence classes would the whole relation have?
If you have a bijection between two finite sets $A$ and $B$ then they have the same number of elements. The converse is also true.
So if $A \sim \{1,2,3,4,5\}$ (that is there exists a bijection from $A$ to $\{1,2,3,4,5\}$) then $A$ is any subset of $Q$ containing $5$ elements.
For any subset of size $0<k \leq 26$ there is a equivalence class containing all such $k$-subsets. So there are $26$ equivalence classes.