Giant Massive Post it is, then. Let's start with the old Forum Knowledge Bases: Ulanopo's Knowledge Base, and Mini KB: Chat Restrictions.
Please understand that both of those represent information gathered over a large period of time, by people who are far cleverer than I. It's my hope to do their efforts justice here. As such, I want to go ahead and state that any failing in the following post is due to my misunderstanding or inability to clearly express the idea, rather than in their work.
Part the First: The Reporting System
The first thing to know about the reporting system is that it doesn't care about individual reports. Instead, it looks at the number of games that you've received reports in. As such, it doesn't much matter how many reports you receive in any particular game, since whether it's 1 report or 9 reports, it's still just one game.
It takes some threshold ratio of reported games to trigger a case. Previously (while the Tribunal was active), this was between 12-25%. There've been no statements about changes to the reporting system, so it's fairly safe to assume that these numbers are still relevant. Once a player has been reported in enough games (relative to the total games that player has played), a case is built against them.
Addendum
A sub-system of the reporting system is the in-client pop-up warning. These warnings appear when a player receives a spike in reported games, relative to his "norm." For example, if you normally receive reports only 1 game out of every 100, but today were reported in 3 out of your 5 games, the warning would be triggered. There's two basic ways a player won't see a warning: rarely ever being reported (keeping your nose clean, basically), or having a very high "norm." In the latter case, the pop-up will fail to trigger, because it cannot detect any spikes, or at least not before a case is built.
Another thing to note about these warnings is that they don't show up immediately. There's a built-in delay intended to obfuscate the specific game that triggered it. It's unknown how long that delay is (though I suspect it's at least slightly variable), or even whether it's tied to the clock or to games played (or some combination).
Part the Second: The Review
This is where things get hazier, unfortunately. There's not a lot of details available for players to see pertaining to the way reviews are handled currently. We do know there are two different review systems (though there may be connections or dependencies between them).
When a case is built by the reporting system, it's compiled and sent to Riot for review. Some cases are checked over by a fairly sophisticated chat-detection algorithm, looking for various indications of negative language and verbal behaviors. If no such indications are found, the case is dropped or Pardoned. If they are, a series of chat restrictions are issued. I suspect that if the algorithm detects sufficiently negative behaviors, it escalates the case for manual review (because that's how I'd do it), but I'm not sure if that's accurate. Regardless, I do know that at least some of the cases examined by the algorithm undergo a second review by a Rioter, to test accuracy (much like the quality checks performed on cars and condoms).
Other cases are sent directly to Rioters. I suspect that this is what happens when the reports are for things other than Verbal Abuse and possibly Negative Attitude. These (and those that may be escalated by the algorithm) are judged in much the same way as the Tribunal handled it, though with far greater access to information than Tribunal judges had. Cases against players determined to be innocent are Pardoned, as before. Those judged guilty, however, are issued 2-week suspensions or Ranked Restrictions. The suspensions appear to be issued in waves, rather than as they come.
It's important to note that cases are judged based on what behavior is found, regardless of what the reports say. For example, if a player is repeatedly reported for Negative Attitude, but the review indicates that he was heavily trolling his team - using Anivia's wall to block them from escaping or making kills, perhaps - the case is still punishable.
Part the Third: The Feedback
To my mind, this is where Riot's current largest failing is. While the Tribunal was active, a "finalized" copy of the case was automatically emailed to a punished player (well, a link to it was, anyway). This allowed for incredibly clear and rapid feedback regarding why the player received a sanction. Unfortunately, when the Tribunal went down, this system went with it.
Currently, chat logs - and other information, I suspect - relating to bans can be obtained by submitting Support tickets on the matter. That policy, however, does not extend to chat restrictions or ranked restrictions at this time. Which, unfortunately, means that players who have received chat restrictions or ranked restrictions have little to no feedback regarding their punishments. Which sucks. I believe that the reason for this is two-fold.
First, the system was intended to continue reviewing the player as their restrictions neared completion, to determine whether or not they had reformed sufficiently. The reissuing of restrictions was, I suspect, believed to be a powerful form of feedback. Unfortunately, the lack of clarity on this seems to have negated whatever benefit it may have had (that, and the buggy nature of how it originally sent out restrictions, but that's supposed to be fixed now).
Secondly, since Chat Restrictions are by far the most common sanction - and manually building sterilized chat logs is quite a chore - it seems likely that the policy was implemented to cut down on overloading the Support personnel. Having seen how overwhelmed they become from ban waves, I can only imagine how much worse it would be on a day-to-day basis if they also needed to respond to chat restrictions with the same clarity.
All of that said, I don't believe that this feedback approach has been very effective or successful, and while I think chat restrictions and ranked restrictions have been shown to be quite effective as sanctions, I'm very eager for the Tribunal's return. Because in terms of behavior education, consistent and clear feedback is the single most important thing. And since the loss of the Reform Card system, I feel that Riot's fallen behind on that.
