Secret Projects of course aren't really secret, since everyone receives a message when someone starts, nears completion of, or finishes one, and you can check the secret projects' menu to see who is currently researching or has already completed which ones.
A Secret Project is a project you can create in any base, to gain a bonus. Only the first faction that completes it, can get it. You can change from one secret project to another in your build menu, without losing any minerals.
You can move supply crawlers to a base producing a secret project, and use the entire mineral cost of the crawlers to help with the project. So in essence, you can have several bases working on the same project, with one actually producing it, and the rest producing supply crawlers to send over to feed it.
If you receive a message that your enemy is almost done, then you can take supply crawlers you were using around your base for resource gathering, and use them to help rush completion of the project.
Many Secret Projects include a cutscene with pictures and audio, and sometimes animation.
Quotes (SMAC) has a list of some of what is said by different people when secret projects are completed.
For a list of the secret projects, see List of Secret Projects (SMAC).