In general, 32-bit applications will run under a 64-bit Windows (This is technically called WOW64 - Windows On Windows). This applies to all 64-bit Windows version to date, including Server.
WOW64 processes are marked in task manager with *32, For example: chrome.exe *32. Sysinternals' Process Explorer has a separate column for this: Image Type (64 vs 32-bit).
If the app has components hosted inside other processes, then those components must accommodate processes they're hosted in. Examples:
- Shell extensions are hosted in explorer.exe, whose bitness matches the OS' bitness. Therefore, you should compile the extension in both 32- and 64-bit, and register the one matching the OS' bitness during installation.
- Kernel-mode driver are hosted in the Kernel, whose bitness also matches the OS' bitness. So, the above applies.
- Winsock LSPs (Layered Service Providers) are hosted in all processes, both 64-bit (native) and 32-bit (WOW64). Therefore, you should compile the LSP in both 32- and 64-bit, and register both in their respective catalogs during installation.
There are also considerations regarding file redirection (System32 / SysWOW64 / SysNative) and registry redirection (Wow6432Node), which I will not go into.