I was going to make something like this, but I'm a lazy bum. Parts of this are from memory, so please don't hang me if I missed something.
Equipment - If you're running a tournament you should already have an idea of what kind of equipment you need. Obviously you need a setup for whatever game(s) you're running, more if it's a big enough tournament and you want to finish on time. A way to record matches if you want to is, again, obvious. Every tournament bracket program that I've looked at has had flaws. Tio is horrible and should be burned to the ground. Tournament Maker is ok for 16-man brackets, but iirc there's a problem with the 32-man double-elim bracket. I don't recall others I've looked at off hand, but every one I looked at (which isn't all of them) had a problem somewhere.
Location - Can be anywhere that you can get people and equipment. Everything else is personal preference.
Rules - Rules are really simple. Just make the least amount of rules possible while covering every detail. Always have a reason for your rules, and make sure your reason isn't dumb. Don't ban anything in the game(/play) that isn't broken*. For some random tournaments it's ok to have otherwise dumb rules like banning some random characters/attacks, but not for serious ones. Don't just copy/paste some other tournament's rules, as from what I've seen most of them are crap. This also has the problem of you not knowing why the rules are there. *What's broken and what isn't is subjective. Usually, if game play revolves around solely one thing/character/technique, it's usually broken. Bugs and glitches will always be a touchy subject, but the usual consensus is if it isn't broken and doesn't cause any major problems(like freezing the game), they're allowed.
Rewards - Can be anything. Most tournaments do 70/20/10 pot split, a bunch do 60/30/10, some do winner take all(usually for smaller ones). Make sure it's something your participants will enjoy, like money. As far as ranbat points, again, it can be anything. The easiest is to just give every place a set amount of points(first gets 10, second gets 7, etc.). One thing you should do is either have it so points diminish over time or have a complete reset where everyone starts back at square one at various time intervals. This is so people who join late, or even just miss one event, will still have a chance to either catch up or can join in and compete on equal footing after the next reset.
Other things - As miz said, keep everyone informed about what's going on as much as possible. Be clear and concise, don't let there be any confusion, whether it's in the rules or in directing the tournament. For example, don't just tell four people that they play, tell people who they're playing, where they're playing, and what game they're playing if necessary. Comfort is an issue that is generally ignored. Having space to move around is fantastic. Having chairs for everyone to sit in when watching is great. While these things aren't exactly part of the tournament, they are a part of the event and can enhance or degrade the experience. Most players are used to playing in bad conditions, but if it can be helped try to make them not so bad.
Tournament Format - In USA, double elim is the standard for most fighting games. There are two other formats that I'll touch briefly on first.
Round Robin: Everyone plays everyone once, you count up wins, whoever has the most wins the tournament. Note that once can mean a set of games, and whoever wins the set gets the point. This is the longest format as well as the most accurate. One drawback of this is it can potentially lead to ties, even for first place. Often times the round robin format is modified, either involving a simple tie-breaker rule(either whoever won between the two wins or they play again), having different points won depending on by how much you win a match, having everyone just play a few times instead of just one, and many other ways.
Single Elimination: A very simple and crude format. It's the shortest way to get a winner and unlike round robin it guarantees there will be no tie for first, however it is also the least accurate. While first place is always correct(barring upsets, which elimination style formats handle worse than round robin style formats), the ranks after that are not nearly as accurate as they could be. While getting second in a single elim tournament doesn't mean nothing, it doesn't mean nearly as much as getting second in almost any other format. The idea is simple, if you lose you're out of the tournament, and if you win you play someone else until there's no-one left to play. A bracket is used almost exclusively as a means of showing who plays/played who.
Another issue with single elimination is seeding, which is basically just your place on the bracket. Your seed determines who you play, and thus the difference of playing someone good and someone bad. For most tournaments seeding needs to be random, due to having no prior information. You can't simply say "this person is better so they get a better seed", or you may as well not have the tournament and just post a results thread without even playing. You can't just go by so and so ranking(like another random tournament), that isn't fair for people who didn't make that tournament. If you really don't like random what you can do is have a ranbat, which is basically just a series of events where you either get some form of points or ranking and seed based off of that. You can either seed by rank a certain amount of people(like top 4/8/whatever), or everyone. This needs to be announced in advance of the first tournament in the series so everyone knows what to expect. This basically turns each tournament into a smaller part of a larger event, whereas the whole series is essentially one big tournament with each individual tournament being the metaphorical matches of the big tournament.
Single Elimination also has a concept not present in round robin, which is byes. Single elimination is only a 'complete bracket' if you have a power of 2 number of people(2, 4, 8, 16, etc.). If you have 15, one person gets a bye in the first round, which essentially means they win without having to play. All byes will be eliminated in the first round; no one ever gets two byes. If you have 9 people, 7 get a bye in the first round, leaving two people to play for a spot in a virtual 8-man tournament. This isn't a big advantage for placement, especially when you consider they could have just played a horrible player instead for an easy win since it's always only the first round. In a very small tournament it can be an advantage(or a disadvantage) for external reasons(like not getting a chance to warm up before an important match), but this doesn't change that the best, barring any upsets, will always win. (Statistically getting the bye is an advantage just because it's one less chance for you to get upset, but again if the tournament is of any reasonable size it isn't likely to change the results past the last couple of places which aren't accurate to begin with)
Double Elimination is exactly the same as single elimination, except instead of being out of the tournament when you lose once, you have to lose twice. This means that there is two brackets, one for people who haven't lost, and one for people who have lost once. The winner's bracket works exactly the same as in single elim, but the loser's is tricky and I've seen many people, and brackets, mess up on it. The basic idea is that people will be moved into the losers bracket so that they enter on a level that keeps an even number of people in winners and losers and so that they are less likely to run into the same person twice(more on this later). When there is one person left in winner's, and one in loser's, they play each other but the one in loser's has to win twice due to it being the person in winner's first loss and you need to lose twice to be out. This tournament takes only twice as long(approx., slightly longer) as single elimination, but gets much better accuracy beyond first place. In double elim, first and second are always correct, and the next few places have a high probability of being correct too. As you get further from first the accuracy declines, however it is much more accurate than single elim. Getting half way through a double elim tournament probably means you're about better than half the people there, whereas getting half way through a single elim tournament doesn't mean much. If you have a consistent group of players and have ranbats and seed by rank the accuracy of the middle places goes up a fair amount. As usual, this also depends on no upsets and due to the somewhat random nature of the lower end the accuracy bump isn't likely to be noticeable in real-world situations, although it can help to nullify one or two extremes.
One of the things I've seen many people, and bracket programs, make a mistake on is part of the loser's bracket. It's possible for two people to play each other twice, which is commonly called double jeopardy. This is normal. A proper bracket will spread people out to avoid this as much as possible. It also avoids similar situations, such as if A beat B, A beat C, B and C playing each other in losers. I've seen people both not separate people in losers properly which creates many of these situations, which in turn lowers the accuracy of third- places, as well as move people around in the brackets to places they shouldn't go, which while avoids direct double jeopardy(only temporarily), it creates similar situations to A>B, A>C, B vs C, which is just as bad as far as final placement goes. Even if you always move people who played each other away from each other, it doesn't even improve third place's accuracy(it shifts the accuracy around for some places though, some become more accurate some less). Just don't do it.
One final note, and this is in regards to the fighting game community specifically as it doesn't apply to all communities, is that many tournaments choose to seed by location, meaning you get separated from your friends/people you play a lot so you play other people in tournament. Firstly, this isn't something that should be done in serious play, however there is a grey area here. It does suck to travel to a tournament only to play your friends, but the tournament doesn't care about that. I don't want to get into the intricate details of it here, but seeding by location may have an effect on the results, however due to how the community currently is, many, myself included, feel the potential accuracy loss is a fair price to pay for not playing your friends early on in a tournament. One of the things that you can't do is seed by both location and ranbat. As an example, lets say my neighbor and I go to a ranbat. My neighbor has 4000 points, the most out of anyone there. I have 3 for getting fifth place when there were five entries on a rainy day. I now suddenly get to be on the opposite side
of the bracket from the best person there just because we're neighbors. My suggestion is to use ranbat poitns if you have a regular group, and location if you don't. I also want to reiterate what I said before about either having points decay or reset so someone doesn't get to accumulate so much that they can never be passed.
As an example of a tournament system, I'm going to use a rough outline of the NFL. First you start with a heavily modded round robin portion. All of the teams are separated into 6 round robin groups. While they play people not in their group, they are basically only competing against people in their group. They play a bunch of games, and whoever has the highest win%(/wins for playing the same amount of games) in their group gets to go to a single elimination bracket. There are also wildcards, which are basically runner-ups that get in. The single elim part is seeded based on how well you do in the round robin part, basically ranbat points. The higher ranked teams get byes while the wildcards need to play an extra game. At the end of it all you have a winner, and then next tournament(/year) it resets. The team that wins the super bowl doesn't get an advantage next season, it's a fresh start and new teams can enter on the same level as everyone else.
Finally, I want to say don't play favorites. The (insert good nfl team here) doesn't get a bye in their path to the super bowl because you like them, they get the bye because they won more games during the season than everyone else. No one gets an advantage at the start, winning/losing during the event determines if you get/don't get an advantage. If you use a ranbat system, you get a good seed because you're doing good in the ranbat, likewise if you're doing bad you get a bad seed, it doesn't(shouldn't) matter if you're liked or not, how good you actually are(if you are then start winning instead of begging for a seed), it doesn't matter how good you did before the points were reset, it matters how good you do now.
And lets add a link to some good brackets. Speaking of which having images with brackets would make things a lot easier to understand.