Text files are great for a server application (you did say Windows 2003). You should have a separate log file for each server application, the location is really a matter of convention to agree with administrators. E.g. for ASP.NET apps I've often seen them placed on a separate disk from the application under a folder structure that mimics the virtual directory structure.
For client apps, one disadvantage of text files is that a user may start multiple copies of your application (unless you've taken specific steps to prevent this). So you have the problem of contention if multiple instances attempt to write to the same log file. For this reason I would always prefer the Windows Event Log for client apps. One caveat is that you need to be an administrator to create an event log - this can be done e.g. by the setup package.
If you do use a file, I'd suggest using the folder Environment.SpecialFolder.LocalApplicationData rather than SpecialFolder.ApplicationData as suggested by others. LocalApplicationData is on the local disk: you don't want network problems to stop you from logging when the user has a roaming profile. For a WinForms application, use Application.LocalUserAppDataPath.
In either case, I would use a configuration file to decide where to log, so that you can easily change it. E.g. if you use Log4Net or a similar framework, you can easily configure whether to log to a text file, event log, both or elsewhere (e.g. a database) without changing your app.