If you grip the tip of your pick so that when you strike the string it catches your thumb, you will hear a harmonic generated over the initial note. This is called a Pinch Harmonic, a type of artificial harmonic. This is a common technique for lead guitarists and while it sounds great on an electric guitar you can also do this on an acoustic. It simply takes a little more practice.
You can also do it without a pick. When you strike the string with your finger or thumb, catch your nail on the string. This takes a little more practice, but like most things it’s amazing how easy it seems once you know how to do it.
Amazingly, harmonics can also be used to tune your guitar. Find out how in the article Tuning with harmonics.