In modern times there are various types of pokéball. The major manufacturers create the basic Poké Balls, as well as Great Balls, Ultra Balls, and of course the rare Master Balls, each type more powerful than the last, and therefore more likely to succeed in capturing wild pokémon. The last, most powerful, type is available almost exclusively to those who have beaten the Pokémon League elite, and to pokémon researchers, although they can also be won in contests. The chances of this are probably one in several million, like a lottery. (There also may be a black market for cloned Master Balls.) Master Balls are the worst of the lot, because they entirely eliminate the "sport" factor. Pokémon have absolutely no chance against them, whether they have been weakened at all or not. (One would think that those trainers who are capable of obtaining such a ball legitimately would have no need of one- unless perhaps they are determined to capture a Legendary Pokémon at all costs.)
In addition to those four types, there are several specialty types available only from Pokéball Master Kurt of Azalea Town, in Johto. He creates them from different types of apricorn (of different colors), which apparently people used to use more frequently long ago, before the creation of modern mass-produced pokéballs. These include the Heavy Ball (created from black apricorns), for capturing heavy pokémon; the Lure Ball (blue apricorns), for capturing water types; the Friend Ball (green), which makes captured pokémon more friendly toward you (tell me that doesn't involve mind-manipulation!); the Love Ball (pink), which captures pokémon of the gender opposite of the pokémon who's fighting for you against the wild pokémon; the Level Ball (red), which captures pokémon of a lower level than the one fighting for you; the Fast Ball (white), which captures pokémon that try to flee from battle (this is another especially insidious type of pokéball); and the Moon Ball (yellow), for catching pokémon which evolve with the Moon Stone. ...I have also heard of a Stun Ball, but I have no idea what it would do, nor am I convinced that it even actually exists.
North of Goldenrod City in Johto is the National Park, where certain days of the week they hold bug catching contests. For this contest trainers are given special Park Balls to use. I have no real data on what's so special or different about them, though I have heard that Ash Ketchum simply handed a Park Ball containing a Beedrill he'd caught, to another trainer named Casey. This would seem to go against the previous evidence that pokémon can only be traded via machines, although it might also be because it was a gift, not a trade, rather than having anything to do with the type of ball used. ...I've heard conflicting reports that in the contest each trainer may be given 20 Park Balls, or just one. If it is just one, it may be that you can throw an occupied ball at different pokémon if you wish to switch which one you capture, thinking the new one might stand a better chance of winning the contest.
I have also heard of a similar contest somewhere in Johto, involving the catching of water pokémon, or specifically Seaking, using Lake Balls. Presumably these would be quite similar to Park Balls, although I would assume there might also be some relation to Lure Balls.
Of course, in Kanto's city of Fuscia, there was, until its recent (perhaps temporary) closure, the famous Safari Zone, where one could catch various pokémon using Safari Balls ($500 got you 30 of them). Aside from its being a business rather than a free competition, this situation is somewhat similar to the bug and Seaking catching contests; however, the balls are really no more specialized than ordinary pokéballs, as they are intended to catch many different types of pokémon. Safari Balls may also be purchased at the Safari Zone on Route 121 in Hoenn, near Lilycove City.
Update 7-24-05:
Some newer balls I've heard of, in the Hoenn region, include the Dive Ball, which makes it easier to capture underwater pokémon; the Luxury Ball, which, like the Friend Ball, makes captured pokémon like you more (apparently, the ball is designed to give pokémon a luxurious experience while trapped inside, a claim which bears further research, with potential to possibly answer some of the questions I raised in my earlier pokéballs article); the Nest Ball, which, like the Level Ball, makes it easier to capture weaker pokémon; the Net Ball, which makes it easier to catch both water and bug type pokémon; the Premier Ball, which is used to commemorate special events- they seem to be given out like souvenirs or party favors- but works the same as an ordinary Poké Ball; the Repeat Ball, which makes it easier to catch the same species of pokémon you've previously caught; and the Timer Ball, which is sort of the opposite of a Master Ball, and the most sporting variety of pokéball I've ever heard of- the longer your battle lasts, the easier it is to catch the pokémon you're battling.
Clearly, there is a great deal of specilization in the ever-burgeoning pokéball industry. There must be a reason for all of it. They just keep trying to make capturing pokémon easier and easier. Who knows what other kinds of balls they'll come up with in the future? It is truly disturbing.