Unfortunately, it's a little bit of a crapshoot.
Firstly, in all cases there is an 'active' and a 'secondary' screen - generally the screen with the cursor is the 'active' screen, and you can tell which is the active screen since it will display the dock and menu bar (you can still see the menu bar on the secondary screen, but it's dimmed to show it's secondary status).
Generally, when you launch an application, it should open on the active screen.
So, in part, which screen an app opens on depends on how you launch it. If you launch It from the dock then, by definition, you're launching on the active screen and that's where the app should show.
If you launch via double-clicking the app icon (or a document), the same thing applies.
If you launch via Spotlight, then it should open on whatever screen Spotlight is currently focussed on
However, there are exceptions.
Some apps don't honor that and always open on the same screen. The only recourse here is a note to the developers.
Some apps also record their previous setup and restore that - for example, an app may record which windows were open when you last quit and restore those, regardless of which screen is 'active'. In Safari's case there are preference settings which offer some control over which way it plays here.
The short version, though, is that there is not a single answer to this question. There is a general default, with per-application support or override for honoring that default, with inconsistent per-app controls over what it should do.