jmc

Since each person shakes hands with each other person, every pair of people will shake hands once. So, 78 represents the number of pairs, which we can count as $\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ where $n$ is the number of people at the party. So, $n(n-1)=2\cdot 78=2\cdot 6\cdot 13=12\cdot 13$. So, $n=13$ gives us $\boxed{13}$ people at the party.