I'm just learning how the court system works, and I'm trying to figure out the scope of influence of each appellate court. If a decision is handed down by, for example, the 9th Circuit court, does that decision affect the other 12 circuits, too, or only the region adjudicated by the 9th Circuit?
It is binding precedent for lower federal courts in the Ninth Circuit. They are required to follow it.
It is persuasive precedent for the other circuits and for state courts. They may be persuaded by the reasoning and will consider the fact that the Ninth Circuit held as it did to be one factor in their decision-making, but they can make a different decision if they want to.
Some courts are also more persuasive than others, although this is usually not explicitly acknowledged in written opinions. For example, state courts in the Ninth Circuit are likely to give more weight to a Ninth Circuit opinion than they are to give weight to a decision from another circuit. Cases from the Second Circuit are more likely to be persuasive than cases from other circuits, because of its reputation. Cases from the Southern District of New York are more likely to be persuasive than cases from most other non-local District Courts.
There is also a personal reputation function that comes into play when looking at persuasive decisions. Some judges have a better reputation or a better reputation with a particular other judge, and their decisions may be considered more carefully. This is not explicit, but it means a good lawyer will mention the judge's name when a case in support of his position was decided by a well-regarded judge.
Finally, the Ninth Circuit's holding would be persuasive precedent for the Supreme Court. You could write law review articles about this, but to dip one toe in: they may be persuaded by the reasoning and it matters to their function of providing unifying law, so especially during the process of applying for a writ of certiorari, they will care what different circuits have held on an issue and which circuits are going which way. But they are not bound by the circuit courts, and will overturn all of the circuits if they think that's the right decision. They did that a while ago with a statute about what it meant to use a firearm while committing a crime, for example.