Views on Alternate Database

The Drupal Views module uses the field APIs to figure out where to make the joins between tables. So no, you can't use it on an arbitrary database without writing a Views handler for that database's structure.

I'm not sure about using a secondary database (I assume it's possible, but I don't know the details of how it works with Views). However, it's definitely possible to tell Views about your table structures, by writing a custom module that implements the hook_views_data hook and returns a structured array representing your database schema.

This can be quite simple or rather complicated, depending on the complexity of fields and joins you need.
Look at the demo implementation of hook_views_data in views.api.php for an example (or any of the other implementations within the modules subdirectory, or any other module with a function named something like *_views_data).

You might also check out the Views Schema module, which automates this process to some extent. There's also the Data module, which provides a point-and-click admin UI for the same purpose. However, I found that the Data module has some limitations, so if you are happy working with code you may be better off implementing the necessary hooks directly.

In some cases, it's necessary to also provide new Views handlers by implementing a new subclass. This is slightly more involved.

Thank you all for your comments and explanations. Sounds like I need to patiently learn more and gain some experience. I had read a little about settings.php and had seen that one can specify an alternate database there, but then I did not know how to connect to it from Drupal in a simple way. You all are telling me that there is no simple way.