This is an overview of the different types of arcade joysticks available. Choosing the right joysticks is one of the most important steps when building a MAME Cabinet or refurbishing a classic game.
Before I get started I must explain the difference between 8-Way, 4-Way, and 2-Way joysticks, as this is very important. An 8-way joystick is operated by 4 switches representing up, down, left, and right. When you move an 8-Way joystick diagonally it will activate 2 of the switches and cause diagonal movement in games that support it. While 4-Way joysticks can only trigger a single switch at once, which disallows for diagonal movement (Except in Q*Bert, which has a 4-Way stick mounted at an angle, and allows for only diagonal movement). While 2-Way sticks can only go left-right or up-down, depending on how you mount them. Space Invaders, Galaxian, and many similar games used a 2-Way joystick.
These joysticks do not universally mix and match. 4-Way games do not play correctly with an 8-Way joystick. The only option for a 4-Way game is a 4-Way stick, because the games simply cannot handle the diagonals being pressed. . Many classic games use 4-Way sticks, such as Pac-Man, Dig Dug, and Donkey-Kong.
Games that use 2-Way sticks (early space games for the most part), will play equally well with any flavor of stick, 2-Way, 4-Way, or 8-Way. The reason being is that any joystick can move left-right properly.
Now most games use 8-Way sticks. The majority of them will play with a 4-way stick, but this ruins many games (like newer space shooters, fighters, etc). There is one exception, that is puzzle games. Games like Puzzle Bobble, Uo Poko, and Money Puzzle Exchanger will play perfectly well with a 4-way joystick. You can't even tell the difference
Why would you want to select the wrong joystick anyway? Because you may have to make some compromises on the controls if you are building a MAME Cabinet, a quick switch JAMMA cabinet, or any other multi-game machine. No control can play every game, so you must select carefully so you can play your favorites.
Universal Joystick (Microswitch)
These are your standard modern day joysticks. They are cheap, and produced under many different names. They can be switched from 8-Way to 4-Way action simply by turning over a little part on the underside. These controls have a kind of an exact feel to them, which is perfect for fighting and puzzle games. They also make a distinctive click with each movement (this is nearly impossible to hear in a noisy arcade, but it is quite clear in your home).
Universal Joystick (Leaf Switch)
These are similar in appearance to the joystick above. But they use leaf switches instead of microswitches. This makes the movement of the stick more silent, and gives it a decidedly different feel. Many older games came with these sticks. All leaf switch joysticks require some adjustment from time to time (as the leaves get stuck or bent too far, etc).
Digital Rotary Joystick
The rotary joystick is a customized version of the Universal Joystick. It features 8-way action, along with a rotating stick that has 12 different positions. This is a replacement part for Ikari Warriors although several other games used this stick as well. It is perfectly usable for other games, but costs nearly 4 times as much as a standard joystick. This one is only recommended for those who need it for a specific game.
Optical Rotary Joystick
This is just like the Digital Rotary Joystick with the exception that the rotating stick is analog in its rotation. This allows the joystick to also function as a spinner. This is officially a replacement part for 50 Caliber, but it would be useful for a multi-game project as well.
Solid State Joystick
These are sold under several brand names (Happ Controls calls theirs the "Perfect 360"). These sticks can be either 4-Way or 8-Way, which is user selectable. But they have no switches at all. Making them maintenance-free and much more silent. They are fairly expensive, but are well worth the price, simply for the easy 8-Way to 4-Way switching.
Fire Button Joystick
These are Universal Joysticks that have a fire button mounted on top. Very few games use these controls normally. With the exception of cocktail tables, which would use them fairly frequently. I would stay away from these, unless you really like the design.
Digital Trigger Joystick
These are 8-Way sticks that look just like a flight stick, with 1 to 3 buttons mounted on top. Not many games used these. Those that did are easily controlled with universal joysticks (which are much cheaper).
The arcade analog joystick looks like a flight stick. With one or more trigger buttons. The analog stick has a large range of movement, instead of just a couple of distinct positions. These are very expensive, but they are the only way to control some games properly.
These have only ever been used in a handful of games. These sticks have 16 distinct directions that can each register at three different intensitys. This combined with the center position makes for 49 distinct positions. These are only used in Gauntlet, Blitz, Blaster, and the sequels to those games. This type of stick was designed as an alternative to the analog joystick, but without the maintenance and high rate of failure that most analog sticks have.
Feel free to send me a /msg if I have missed any arcade joystick types.