I have taken a theme and converted it to a component and saved it. It then shows in the component library. However, when I close the program, open it in another file, and try to access the component it has disappeared. If I go back and load the theme that I made the component from, the component displays.

How can I save the component so it shows in the library, no matter what file I open up?

Make sure you are using the same Framework. If you created a component for Foundation for example, that component can only be used for Foundation projects. It sounds like you switched to another framework (like Bootstrap) which will not show that component.

Thanks. That is exactly what happened.
Is there a way to change a Foundation them to a Bootstrap one so I can use it in my Bootstrap project?

Also, can I find out what method (Bootstrap, Foundation, etc.) a project is made in.

No. You cannot convert projects or components between frameworks.

You can always see what framework is used by going to the Layout pane.

What framework do you recommend for projects? Is one favored and/or easier than the others? I'm not an experienced programmers, but can code HTML and use the Responsive Site designer ok.