The "service salt" (described as "a sitewide static salt" in the question you cite, and also sometimes called "pepper" in crypto literature) is simply a secret string which is fed to the password hashing algorithm along with the password and the per-user unique salt.
The point of having a "service salt" like that is that, unlike the per-user salt values, the service salt is not stored in the database but somewhere else (typically in a configuration file, or hard-coded into the application). Thus, it protects the passwords against attacks that only compromise the database but don't allow the attacker to access the app configuration. With modern web apps, such an attack scenario is not as unlikely as it might seem; for example, a simple SQL injection attack would often fit this scenario.
One detail to keep in mind is that, unlike the per-user salt, which just needs to be unique and not too easily predictable, the "pepper" actually has to contain a substantial amount of entropy (say, 128 bits) and must be kept secret for it to be of any use.
In any case, including such a secret constant in the password hash calculation is pretty easy, so there's very little reason not to do it: even if your password hashing algorithm doesn't explicitly support it, you can just, say, append the "pepper" to each password.