Rather, in our context here, the “stick” is actually a controller, intended to emulate the feel of arcade input by using arcade-like parts (or even better, actual arcade parts).
The stick is composed of a joystick and buttons, and with the exception of special cabinets like Buriki One, you move around with the joystick (see directional input), and perform other actions with the buttons.
The gamepad is the main alternative; as you might expect, the buttons and directional input work in the same manner, but since the pad is controlled with your fingertips instead of your hands, there are going to be some that find this harder or easier.
Of course, the arcade is generally what people consider when it comes to the snootier tournaments (e.g., Evo, while boasting a wonderfully high level of play, still runs on consoles) and showing the general public who’s best at the game, so it’s best to practice home-use ports with arcade-style controls if you want to be better at the arcade. Or perhaps, if go to Japan and play against the best, you have no choice but to practice with arcade-style controls.
The hard part, really, is acquiring arcade-style controls. They are not difficult to find per se, but at the time of this writing, you would probably not find them in any old brick-and-mortar game store anywhere in America (though occasionally a new game will come out, like Tekken 5 or Street Fighter IV, that actually advertises a decent set; it’s a good idea to take advantage of golden opportunities like these to get a decent stick for less money). The real problem is price. You pay a premium for these specialty precision instruments. You pay a premium for better parts (like if you want it lighter, or to light up, or if you want it quiet because you live in a cramped apartment complex, etc.), and depending on whether you get it custom-made or consumer-grade, you may pay a premium for labor. These things add up.
Probably the cheapest way to get a good arcade-style controller is to build it yourself (or be a very good friend of someone who does). Unfortunately, it does take some skill to construct these, and you still need to get the parts themselves (which is beyond the scope of this article by far; I recommend you check out the Shoryuken.com “Tech Talk” forum for more information on parts than you could ask for).
The next-cheapest way (generally) is to get a consumer-grade solution. You can often (though not always) modify it yourself if you want to. Read reviews on the model you’re interested in. Don’t get the cheap stuff if you can avoid it, or you will basically get a near-useless piece of hardware. Do look for good deals, though; a quality model like the Hori Real Arcade Pro series is usually somewhere between $100 and $200 USD, but since the HRAP1 and HRAP2 and various related models are PSX compatible, you can actually get adaptors for it to work on the GameCube (and thus the Wii), Xbox 360, PS3, and PC (but do watch out for lag-inducing ones! They are useless because they don’t do their job on time!). Knowing what the reviews say will practically assure you that you are getting a the item you want, and what prices are reasonable. (My only recommendation in this category is the Hori Real Arcade Pro. It is wonderful if you like Japanese parts and layouts and don’t mind that it’s 6 pounds of metal and plastic that is truly difficult to bring to harm; it is not so great if you don’t like some of these properties!).
Finally, you can commission someone else to build your stick for you. This can be cheap or expensive (it depends), the results can be good or bad (it depends), and so on, but depending on who makes it for you, you may be allowed to specify what exactly you want (materials, specific-brand parts, artwork, what kind of finish, etc.), and you may be getting the best possible sticks there are (one of the reasons being that what you end up with is yours and nobody else’s, if no other reason).