For the most part, if your application purely uses things built-in to Delphi, then you shouldn't have any problems at all. Delphi automatically uses the Windows API to be backwards-compatible. Therefore, if you have a very simple application using only the things which Delphi provides to you, then chances are your app should work in any (current) version of Windows.
On the other hand, if your app makes use of any direct Windows API calls, or if it uses other third-party libraries (such as ones requiring additional DLL's), then you may be in for an upgrade.
The Windows controls for a forms application are extremely standard, and adapt to all versions of Windows - and even can even automatically style to whatever version of Windows you're using. However, if you have any third-party custom controls, or anything which access the Windows API directly, then you will need to make sure those calls are compatible.
When using the Windows API, it's safe to assume that most of the calls are backwards compatible. Microsoft is very good at this. Again, if using the standard components/controls available in Delphi, and not directly accessing the Windows API, then you should be safe.
When it comes to 32/64bit applications... Every version of Windows supports an emulator of the prior bit-wise version. So, for example, all 64bit versions of Windows support running 32bit applications. On that same note, all 32bit versions of Windows support running 16bit applications. This support is through an emulator (WOW64) allowing even further support of backwards-compatibility.
In the end, as long as your application is purely using what Delphi makes available to you, and you aren't importing any additional third-party libraries, then you should be safe to go. I would say a 98% guarantee. The other 2% has to do with whether your app is using anything which relies on particular 32bit libraries.
When it comes to third-party libraries, the rule of thumb is to just make sure the library vendor claims they support the version of Delphi you're using. That still doesn't always mean it will work with 64bit though. For example, things which directly use drivers will also have to be sure to support either 32bit or 64bit drivers.