Let's take a quick look at several generation methods, and how they are employed, before we discuss how each of them impacts the character generation. I speak from more than a decade of experience here.
Roll for Stats
Rolling dice to generate a spread of numbers that represents the stats of the character has been a classic since the earliest days of D&D.
In some games, such as The Dark Eye 3rd Edition, you just roll your stats, choose a class that your statistics fulfilled the requirements for, and then copy the skill list over, you're done. In other games, such as D&D 3.5, after statistics and class choice, you need to make some more choices, such as distributing your skill levels.
The general benefit of Random statistics is the relative speed at which you can generate characters. It also allows characters to be very different from one another. The biggest downside is the randomness itself: you might end with very much sub-par characters just as much as too good ones.
In some games, you also might end with an array that can not qualify for the class you wanted, for example in TDE3, if the class requires a 16 in one statistic and the highest you ended with is a 15, you can not take that class at all and have to choose a different.
To alleviate the problem of characters being too different, common houserules address when you can reroll stats (e.g. "if your total is lower than X"), extra rolls to choose from or even communally creating the array of stats for all characters.
So, the lesson here is that randomness can be quick, but severely limit the choices and can lead to disparate characters.
The counterpart to the random creation of statistics is a standardized array. Everybody gets the same numbers, only distributing them differently. You can not change that array. For example, Pathfinder 1st Edition did suggest this method and (iirc) used it for organized play.
This method is intended to result in very comparable characters, and indeed manages to do so by pretty much robbing them of any variation in this step: everybody gets the same array, and even if you add some kind of race modifier, the resulting characters are very similar and easy to compare. By designing the game properly, you can end up with all characters having stats in a somewhat narrow band of statistics. This can make encounter design much easier for DMs in the long game. As the standard array + modifiers are easy to verify, character generation can happen separately from the actual game session without much trouble.
The downside is, that the array might not be perfect for a min-maxed character, and usually, the array contains a very distinct "dump stat". This could be alleviated by offering multiple arrays to choose from that are deemed equivalent by the designer, but finding such arrays can be hard.
Point Buy statistics
Going a step further to free generation of the stat array, you could go one step further: You get a number of "Build points" and from that pay for your statistics according to a specific key. Again, Pathfinder 1st edition did offer this method together with rolling for stats and the standard array in their core book, making it a very well-known method.
The main benefit is, that players can tinker with their statistics to their heart's desire, and if properly done, high statistics have a well-designed cost so that someone with an exemplar statistic needs some lower one(s), and that the resulting statistics end in a specific band.
A side effect is, that GMs usually don't need to oversee character generation if this is the only part where build points are used, but it might be more time-intensive than the array.
This leads directly to the downside: figuring out the perfect array is time-intensive and can involve a lot of math, and optimized characters will often have extremely similar statistics but for occasional choices.
Standard Array To Buy Statistics
In a twist, let's take a look at the Storyteller and Storytelling system, which combines both standard array and point buy.
You get a standard array for the statistic block of the sheet made up of three categories: Physical, Social, Mind. Your array reflects (7/5/3) free points that can be distributed 1:1 onto the three attributes in each category, and each attribute already starts at 1 usually (some splats might alter this). For example, the physical category is commonly Strength/Body/Dexterity. You are free to distribute those points as you wish, as long as you stay with 5 or fewer points per attribute.
Then, at the end of character generation, you have "freebies" and spend a number of those to add additional points on attributes.
The benefit here is, that characters can be very tailor-made, but due to the system itself, there are some setups that are clearly superior to others, due to how skills later interact with the statistics. For example, Appearance is one of the least used attributes, and notoriously tricky.
The method is usually easy enough for even complete newbies to create a character they can identify with, but masters of the system can create characters that are much more capable than others. Also, these methods need a much more narrow band of statistics to keep characters in any way comparable - the two systems strive to keep the statistics between 1 and 5 in general and 0 to 10 in extremes.
Shadowrun employed a rapidly different approach to the standard array. In 1st to 3rd and 5th editions, the standard character generation method was the priority system. In 4th edition, Priority was a later option.
The SR priority system offers usually 5 categories with priorities A to E, and you choose to assign each category one of the priorities - A was the best, E was the worst. By choosing one or another level for the Attributes or Skill category, you then were given a number of points that you could distribute 1:1 onto the Attributes/Skills respectively. The other categories unlocked species, magic and cash according to their priority. Each species also had a baseline and limit for Attributes with a hard lower limit of 1 for all attributes.
The main benefit of this is, that it speeds up the calculation of characters and characters, speeding up character creation and verification, but some combinations of priorities always were better than others due to the impact. As an offshoot of point buy, most of the things said there also apply.
And now we go off the rails: Why just generate your attributes ("stats") with a point buy system? Games like Legends of the Five Rings 4th Edition, The Dark Eye 4th Edition, and Shadowrun 4th Edition natively generate characters by using a Point buy system where the buy points are not just for the attributes:
- TDE4 has you pay for species, culture, profession, attributes, benefits and flaws from the same pot. Then, calculate your skill baseline by summing up the boni from species, culture, and profession. Then, calculate the skill point pool and use a complex spreadsheet to buy up your skills from the baseline you calculated a step earlier. Due to the nature of the beast, this is extremely detailed but also very complex and very hard to verify at times. Even someone who knows exactly what they do can spend hours to just make a baseline character without computer assistance.
- L5R 4e has you make a couple of choices that determine a baseline, and then you pay for all advances from that with points to gain merits, flaws, and increases on skills and stats all from the same pot. This gives characters a lot of variance, no two "Kakita Bushi" I ever saw were exactly the same sheet, especially because every skill is useful in some way or another.
- Shadowrun 4 natively handed you 400 points and a price list. You paid for your species to get a baseline attribute spread, then paid from the remainder for all the attributes, skills, merits, flaws, and cash you might want on the character. Due to linear costs, this very much benefits one-tick ponies and creates at times very gappy characters. But characters can be extremely different, even if very much optimized.
- The alternative Karma system for Shadowrun 4 replaces the build points with pretty much XP, and the payment structures, especially for higher numbers, are shifted to mirror the in-game progression. The resulting characters are often much more well-rounded than the normal SR 4 Point Buy.
In summary, "Buy Everything" systems are very complex but result in characters that can be extremely fine-tuned to a player's desires. A downside is, that a GM might have trouble following the character generation after the fact if the system is too german (TDE4). A very complex buy-system can also shock new players, and they generally don't work too well with class systems unless you use the class as a baseline to work from.
Abort Chargen for direct play?
In some rare cases, you could abort Chargen and instead go to play the characters in an incomplete state. Of the systems mentioned in this category, only two actually could facilitate that:
- L5R 4E, in theory, leaves you with a rudimentary, playable character right after choosing the family and school, at which point you'd begin to play with all rings on 2, two attributes that received a raise from family and school respectively (and possibly a ring that is 3 as a result), 7 skills from the school and 40 XP that can be spent just like experience in game later. Just, all characters of the same school are the same in this state.
- SR4 using Karma Generation usually grants 750 Karma (=XP), which is the same as in the game. Only a select number of choices (species, buying of certain merits such as magician, setting the attributes) actually are fully required to get to a state you could play, and then spend the rest of the balance ad hoc in game, but that can very easily lead to a very messy game.
No More Numbers
Now, some games, such as Fate and Powered by the Apocalypse engines, (almost) totally abandoned numerical values for character generation. Instead, they use descriptors or keywords (Fate) or playbooks (PbtA) where you choose one of several sets of pregenerated stats. In effect, often these are really fast to generate characters with (find good keywords or choose a playbook, make one or two choices and you are done), but those are made from an entirely different than the other, more traditional systems I explained above.
Freeform Roleplaying - MUSH edition
Something that you might encounter a lot in online RP settings are freeform systems, where you go into character generation with just the equivalent of a pen and paper and your ability to convince the people organizing the game that you are a well defined character. Often, such games have a distinct theme and you explain your traits to whoever would look at your sheet.
For example, take the following wording for an "Athletics" skill:
Physically fit, [the character] is equivalent to a skilled sportsperson. [The character] would have little trouble to pass the physical fitness exam for the military without extra training.
A whole character could be made up of snippets like those, but writing them often uses a lot of time, as you need to make sure that all those people unfamiliar with your character get an idea what you are talking about. Often, games that work with such sheets are totally freeform and could be more described as cooperative storytelling than traditional RPGs though.
Making it up on the fly!
In a game that shuns traditional sheets, as VLAZ correctly noted, you could totally not generate the character at all before the game starts, but discover or explain things about your character in game, until the sheet is fully filled in. However, this is not exactly an easy thing to design your game around.
As someone who started RPGs with a couple of rounds of The Dark Eye 3, I learned to love the point-buy-everything system of The Dark Eye 4/4.1 for its intricacy and deep character building. The system is complicated and many players use a computer assistant tool to keep calculations straight, but there is something that is neat about how they set it up.
As horizons stretched, I gravitated over Shadowrun 3 and 2 to 4 and 4A, and there to the Karma system, for it would fit my style of making characters best. I learned to love Legends of the Five Rings 4th Edition as one of my main systems currently not just for the background, but also because it somehow managed to make the whole character generation both carry meaningful choices, create varied characters but also be reasonable fast.
Next to those, I love the Storyteller/ing system from the World of Darkness and Exalted 2.5/3, though the latter two have so many options to choose in some steps of chargen, that making a character is an exercise trying to fight choice paralysis when it comes to charm selection. The former however manages to feed both my detailed character options need as much as being a fast system, without bogging down with decision paralysis by simply limiting the availability of options for later.
While I have played my fair share of D&D 3.5, I found that character generation in Pathfinder 1 was easier for me, mostly for streamlining reasons that were made. While I understand the draw of D&D 5e and I have dislikes for many of the design choices made by WOTC, I must admit that they managed to get Character generation quite well down, though to me, characters made in all those often feel flat and stale to me. At least they do compared to the intricate characters that I craft with other systems.
But that all comes down to personal preference. All systems have their benefits and drawbacks, and personal appreciation is one of them. Choose your stat generation to fit your design goals and make sure it just feels right for the game you write.