There are two approaches to running Windows apps on a Linux box: emulate the Windows API on a Linux box, or provide a Windows environment running under Linux. The first approach enables the app to run in the Linux OS, whereas the second enables the app to run in a Windows OS on top of the Linux operating system.
For the first approach, the best bet is the WINE open source project, which provides a toolkit to port apps to the Linux environment as well as a loader that enables some Windows apps to run unmodified.
The second approach requires a commercial product. The best known is VMWare, which allows Windows to run as a hosted OS on top of Linux. Naturally, it is not free, but it is very well-regarded.
Without knowing more about the specific applications, it's not easy to recommend one approach or the other. The least expensive approach would be to start with WINE and move on to VMWare if WINE doesn't work out.
This was first published in September 2004