Interesting problems that can be solved with algorithms
Depending on the level of sophistication of your intended audience, a simple canonical example is testing whether an input number $n$ is prime. 

A brute force approach is to test, for all $d$ in the range $2,\dotsc n-1$ whether $d$ divides $n$. If you find a $d$ that divides $n$ you can stop and assert that $n$ isn't prime; if you don't find any such $d$, you can assert that $n$ is prime. That'll take about $n$ divisions if $n$ is prime. 

An improvement is to try just those $d\le\sqrt{n}$ (rounding up), since it's easy to see that if $n$ has any divisors, there must be one in that range. This is a big improvement, since it reduces the number of divisions from $n$ to $\sqrt{n}$, more or less.

If you want to expand this, there are even faster ways to test for primality and if your audience is sophisticated enough, you might want to try one or more of them.