I trashed my Aperture preferences, so now I can open Aperture. The dialog box says Welcome to Aperture. Will I still have my old Aperture Library?

Trashing the preferences does not remove the Aperture library. The only problem is, that the location of your Aperture library is stored in the preferences. So now Aperture does no longer know where your Aperture library is located and what it is called.

By default it would have been located in the Pictures folder and called "Aperture Library.aplibrary". And that is where Aperture is looking for a library. Do you remember, where you stored your library? Then select it in the Finder and double-click it to open the library in Aperture. After that will Aperture know again where your library is.

I have looked in my Picture folder and see "Aperture Library", but not "Aperture Library.aplibrary". Are they the same?

Have you set the Finder to hide the filename extensions? Then enable the extensions to be shown in the Finder Preferences (Advanced tab):

You could also have hidden the extensionin the "Get Info" panel for the Aperture Library:

Can I open this new "Aperture Library"? If I do open this new "Aperture LIbrary," will my old photo library be there?

If that library only has been created yesterday, it is unlikely, that it will contain your photos. How large is this "Aperture Library"? If it is very small, only a few MB, then it is probably a default library, that Aperture has created, when it could not find your regular library. But if it has several GB, than it can be your Aperture Library - then double click it to open it and see.

If your old "Aperture Library" is not in your Pictures folder, enter ".aplibrary" in the Spotlight searchfield in the upper right corner of your screen.

Does that turn up more libraries?

and I got the message again that the program was rebuilding when it shut down,

but I try to rebuild nothing happens.

You mean, you are seeing panel with the warning and an option to "rebuild" or "cancel"? What do you mean by "Nothing happens"? Is Aperture hanging, when you press the rebuild button, or is the panel unresponsive?

If I click Repair, then the message goes away, but there is no way of telling if it is Repairing or not.

In that situation should you really try to repair. That may take a while to finish - hours or a day or two for a large library.

When you select "Repair" , does aperture show as "running" in the Dock? With the tiny blue dot besides its icon?

Does the Activity Monitor show, that Aperture is using memory and CPU time? Then let it run for a while.

There is no blue dot next to the Aperture icon.

Did you look directly after presing "repair"? Then Aperture is really not running .

Also How do I check the Activity Monitor?

Activity Monitor is one of the applications that comes with every mac. It is installed in the "Utilities" folder in in the "Applications" folder. Just open this folder and double click the "Activity Monitor" application to launch it.

But if Aperture is not running at all after clicking the "repair" button, the Activity Monitor will not show any progress, I suspect.

Have you tried to run the First Aid tools manually by holding down the ⌥⌘-key combination and selecting "rebuild"?

Try it again and let it run, look at the Activity Monitor.