I don't believe that there are clear "you skip step x" things for most firms. Already for the reason that the organisation has to protect itself against nepotism, favouritism and for compliance reasons. In some firms, you may be invited without a more thorough screening, but that's where the skipping ends.
But, as you rightly said, it helps to give your application an extra bit of attention that can make all the difference. The organisation implicitly trusts people working for it, so a referral will give you a bit of a head start in the interview process. Depending on the level of the referrer (Partner vs. Analyst), that can be more or not so much.
In any organisation that I have seen, there are some key players or power brokers whose judgement or recommendation is highly respected by their colleagues. "If X recommended the guy, I can almost trust that blindly. Her recommendations are always spot on." If your recommendation comes from such a person, especially in smaller companies/offices it may even take your application from a default "NO" to a neutral or even default "YES"
So to sum up: It definitely can't hurt, but there's also no "If this... , then that..."