This can only be answered properly if you ask about a specific app. However, for the vast majority of apps there are three things you need to be aware of:
the app bundle
Inside the Applications folder (or wherever you've moved it to), you will find the app. If you have a backup (eg: with Time Machine) of an older version, simply restore the old version.
Technically, this is the only thing you need to do. But it won't always go smooth, so continue reading.
Almost all apps will write to a file
~/Library/Preferences/com.developer.AppName.plist and some will write to other locations inside
~/Library (a common one is
Depending on what the developer has done, these files may or may not need to be reverted to a backup that you made before upgrading the app (again, Time Machine is probably the best way to revert these files).
Most of the time, it should be fine. Personally I would try using the old version without reverting these files, and if you do run into problems - then revert them.
(note: the paths have changed a bit for apps that use the new sandboxing model... perhaps someone can edit my answer to describe how)
Any document you've edited in a newer version of the app, might not be fully compatible with the old versions. For example Microsoft Word 2003 documents do not always open perfectly in Microsoft Word '98, depending what features you used.