Harmonics are magic-sounding notes created in a slightly different way to a normal fretted note. Here's a very easy beginners lesson on how to play them.
Make sure you download the free The Bass Guitar Resource Book which contains (amongst a lot of other cool stuff!) a fretboard diagram of all the natural harmonics as well as some great chords to play.

All the natural harmonics
Once you've developed the required touch and technique to play a natural harmonic, it's time to work out a few places on the neck that sound good. The Bass Guitar Resource Book has a bunch of chords that sound effective for solo bass, endings, and bass lines.
Here's a video with the chords.
You might then want to think about artificial and tapped harmonics. They're a step or two up, sure, but they sound awesome.
Make sure you listen to the masters playing harmonics. Jaco wrote the book on them...
Making music with various bass techniques is the goal. Once you know how scales and keys fit together, you can use harmonics with chords and scales and create your own tunes.
Here's a lesson on how to do that.
Hi Dan,
I really enjoy your teaching! Does string height have a huge effect on the clarity and tone of the harmonics? Also the bass itself with regard to pickup configuration? I have Fender Jazz and Fender Precision both Mexican with all stick components. Will that have and effect as well?
Thank you,
Scott
Apologies for the mistype. The word “stick” should have been “stock” meaning original.
Don’t worry, I got it!
Thanks very much Scott! Nice of you to say. There probably is an optimum height for clarity and tone but, to be honest, if it’s comfortable for you to play them well then that’s more important. The back pickup of a Fender Jazz is usually going to give you more clarity than a P bass. However, harmonics can sound great on a P bass too! There are plenty of options and it’s really all about what you have at the time, and what you’re hearing. Once you have the technique down you’ll be able to do loads.