1. Key fact: An equivalence relation on a set corresponds exactly to a partition of that set.

2. So, the number of equivalence relations on $\{1,2,3\}$ equals the number of ways to partition the set $\{1,2,3\}$.

3. Let us list all partitions of $\{1,2,3\}$:

   (i) One block: $\{\{1,2,3\}\}$

   (ii) Two blocks: $\{\{1\},\{2,3\}\}$ $\{\{2\},\{1,3\}\}$ $\{\{3\},\{1,2\}\}$

   (iii) Three blocks: $\{\{1\},\{2\},\{3\}\}$

4. Total number of partitions: $1+3+1=5$

5. Therefore, the number of equivalence relations on $\{1,2,3\}$ is $5$

6. Option check:

   • A: $7$ ✗
   • B: $4$ ✗
   • C: $5$ ✓
   • D: $6$ ✗

Hence, the correct answer is C.