Users who want to know if the hardware on their existing computers is adequate to run Windows 7 can use Windows 7 Upgrade Advisor, a free tool from Microsoft, to check. This tool is very easy to install and use, and is available for download here.
Users who want to know if applications running on XP (or Vista) are compatible with Windows 7 can use Microsoft Application Compatibility Toolkit 5.5, a free tool from Microsoft, to check. This tool requires some configuration, and involves client package distribution, data analysis and issue tracking. To learn how to configure the tool and how to use the tool, read resources online at Microsoft and watch these four videos. I find these videos are very helpful, and hope you do too.
The demo application Stock Viewer shown in some of the videos is available for download, thanks to Chris Jackson for sharing it. It works on Windows 7 (32-bit or 64-bit).
Almost all applications running on Vista should just work on Windows 7. If you need to run some applications in XP mode on Windows 7, check out this How-To article.