What I intend to do in this article is look at the very basics of the different dice systems that are out there, and could be considered when developing your own RPG. This list is by no means exhaustive, and may (okay, definitely will) be biased by my own experience and feelings. I will however try to stick to the facts and leave my own experiences until the end of the section.
When looking at dice systems there are essentially two basic models. The linear model and the bell curve model.
The linear model is the kind most popularised by games such as dungeons and dragons and other systems like Chaosium's percentile system.
In these systems, a single dice (or dice type in the percentile system) is rolled which produces a random result along a number line. The success range of this number line can be altered by modifiers, but essentially you always have the same chance to roll any one number on that line.
A classic example of this is the core mechanic of the D20 system, and indeed the resulting Dungeons and Dragons system since 3rd edition. Here, most actions involve the rolling of a single D20, with the result and/or target numbers being increased or reduced by modifiers. The target number (or difficulty class as it is identified in D&D) is set by the situation or obstacle that the character is trying to overcome. In percentile (D100) systems such as Call of Cthulhu or Rolemaster, a similar system is in play, but in these cases the target number is often the character's ability or skill number, again altered by any system modifiers.
Regardless of what the actually mechanics of the system are, the nature of the linear roll mechanic is the same. That being that you have a flat distribution of results across the die range and you can always have quite 'swingy' results. For example, in the D20 system, a natural 20 is always a success (5% chance each roll) and a natural 1 is always a failure (again, 5% chance each roll) and these numbers are just as likely to come up as any other.
Bell Curve Distribution Models
Bell curve normal distribution is a term that may not be familiar to you unless you've already seen some basic statistics in your life, but the concept is essentially simple. In a system where you roll more than one random die and add them together then, on average, you get more results towards the middle range of possible rolls than you do your outliers (i.e. more average, less critical successes or critical failures).
Two good examples of such systems are Dragon Age and Dungeon World which use a 3D6 and 2D6 system respectively. In these core mechanics, you roll the dice as indicated, plus or minus and modifiers and compare it to a target number of some description. While this may seem similar to the D20 style systems above, there is a very significant difference.
If we look at rolling 3D6 for example there are two differences to rolling a single D20. Firstly, the range of 3D6 is 3-18 which is narrower than 1-20 generated on the D20, so the result is always going to be across a smaller range (and therefore a bit more predictable). Secondly, it is made more predictable still by the 'averaging out' that results from rolling multiple additive dice. Essentially, you are more likely to end up with a result around 10-11 (12.5% for either or 25% for both) than you are to hit any of the far outliers like 3 or 18 (0.46%). Compare this to the D20 system where any number has a 5% chance of coming up each time, and then you might start to see how different the gameplay could become.
A 2D6 system has a smaller range (2-12), but the outliers come up more often (e.g. 2.78% chance of a 12). This is an important consideration if you increase the number of dice rolled in your system. For example, if you used 5D6, you would increase your range of results (5-30) and your average (17-18) but your odds of ever getting the big outliers really drops (i.e. only a 0.01% chance of getting a 30). However, due to the way normal distribution works, you still have about the same odds of seeing the average come up. For systems where this could come up more, see the dice pool systems below.
A bit of a quirky system that also falls into this category is the Fate/FUDGE system. Here dice are six sided but have two sides blank, two with a '+' symbol and two with a '-'. These essentially give values of 0, +1 and -1 respectively. In the Fate systems, the standard mechanic is to roll 4Df (or 4 Fate dice) and apply the result to a skill or approach. What this essentially generates is a bell distribution curve that averages around out getting a modifier of zero (23.46% of the time), but ranging from -4 to +4. It's also worth noting that you usually get a result between -2 to +2 (88%) of the time, which is why Fate revolves around +2 modifiers so much. What it essentially boils down to, despite the symbols, is a 4D3 system, hence why I have included it here, and not in the next subsection.
Other Distribution Models
Most specifically in this section I'm going to discuss dice pool systems. These tend to be more complex and less intuitive than your roll over or roll under systems; however, depending on the complexity of the system and the experience of the player, they can be quicker to factor in modifiers.
Dice pool systems such as those used by the White Wolf Storyteller system (and the 1989 Shadowrun system) might need a bit of explaining outside of the bell distribution category as the distribution of the results vary. Here multiple (or rarely a single D10 is rolled), but these generate a secondary result based on what are called successes. In this system, a roll of 8 or above (edition dependent) on each die generates what is called a success, and the number of these successes is what's important.
Again with such systems, modifiers can do things like change the number needed to be called a success, or the number of dice rolled, but essentially, you still have an equal chance of generating a success on any single die, however, he more you roll, the higher your chance of succeeding, and therefore the distribution curve shifts.
These systems can be further complicated by rules for things like botches (e.g. 1s take away from successes and could make failures worse), exploding dice (e.g. a roll of a 10 lets you roll another dice to add to the poll). Therefore, I don't want to get into the statistics of these systems as I want these notes to be as simplistic as possible.
An alternative, but less complex version of a dice pool system uses an additive system rather than the 'successes' style model above. Here they stick more closely to the roll over model, but the number of dice you roll is increased or decreased based around your character's abilities and other modifiers. In some ways these systems could fit into the bell curve distribution model, but I have kept dice pool systems together, as the distribution size shifts significantly with dice pools compared to the other examples in the previous section.
A good example of such a system would be the D6 system popularised by West End Games (most notably for Ghostbusters and Star Wars RPGs). Here players used a number of dice based on characteristics and skills that were rolled, the results added together and compared to target number. Increasing the number of dice rolled in these systems increases the mean result, and the maximum result, but does push the probability of your actual result towards the mean. A small point, but one worth considering.
Other dice systems
There will be many other systems out there that I haven't covered, but for the sake of my sanity as well as your own, I'll only touch briefly on one other variation I have come across. Both savage Worlds and Deadlands RPGs utilise multiple die types in their core mechanic. Simply put, one ability or skill may use 3D6, while another may use 4D4 and so on. I won't spend time on these, as although the use of differing die may add to variation in the results, essentially the systems still follow the basic principles of those outlined above.
By way of getting my own opinions off my chest, I do have experience with many of these systems and strong feelings around some of them. With respect to the general D20 vs. 3D6 vs. Dice Pool systems I have enjoyed them all, although I do prefer D20 or 3D6 models more than dice pool systems. I think it's more that I find messing around with dice a bit tiresome and, to me at least, not what I want to spend time doing in game.
To that end, I always found Deadlands RPG highly irritating. While it may seem cool to use all those many dice you have collected over the years, it was a colossal pain having to round up enough D12s or other dice for different abilities in game. I'm just not sure it adds enough to the game to justify the hassle. I should add that I have never played Savage Worlds, so although the dice system has put me off for now, I wouldn't want to say it would be the same for me as Deadlands.
With respect to the D20 vs. 3D6 systems, I do at times quite like the randomness of the D20 over 3D6. While a normal distribution can be fun, the excitement of going from miserable miss, to critical hit does have its moments and certainly fits in with the pulp high fantasy of games like D&D. However, for Systems like Fate where narration and competence are favoured, then normal distribution works well.
As non-committal as it sounds, it's safe to say that there is no perfect system. Each has its own merits and its own drawbacks, and really it depends on how you want your games to be. As my experience hopefully shows, 'swingy' results can be fun, but normal distribution results will be more predictable. Dice pool systems can be faster, but complexity can slow them down, and inexperienced players may find them less intuitive. It really is up to the referee and players to decide which they prefer. Or even if they want to use dice at all...