heres the basics…
butterfly shape is nice and wide so you can land the gap of the yoyo on the string, these yoyos you typically want to spin as long as possible… you accomplish this by developing a strong and balanced throw, and by maximizing the rim weight, and thus maximum momentum… the momentum keeps the spin going, and the gyroscopic force keeps it stable… minimizing center weight keeps the yoyo light and fast… these properties make the angled shape like the yyf protostar or supernova excellent yoyos
imperial or thin, more circular yoyos are used for looping tricks… to see a great example, just youtube search shinji saito, and you can see how amazing it really is rather than reading my opinion about it…
any ways, this style requires sleeping, but sleep time is not as essential as it is for string tricks… more importantly, the yoyo flips 180 degrees after every loop… so a little less stability makes it easier to accomplish so more center weight and the imperial shape is the result
string trick with two yoyos is best with the butterfly shape
offstring yoyos, typically have a smaller gap, making it easier to respond, sleep time is still important here and there are also yoyos with rubber rims for durability because being off string can occasionally lead to a run away yoyo
and counter weight yoyos should share the same ideals as the string trick yoyos…
that pretty much sums up the major five styles, but another thing to consider is that more advanced players typically dont want the yoyo to come back with a tug, instead a bind return this helps enhance sleep time, and opens the door to tricks that would otherwise snap back to your hand, and believe me, that can hurt…
looping is meant to be responsive
i hope this answers your questions and gives you a nice understanding of how yoyos are supposed to work, and i see yoyojamiszm and studio42 covered grinding, finishes and materials… beyond a certain point, the material dosnt really matter unless you want it to frequently contact while spinning (any kind of grind)
anyway, enjoy the journey yoyoing takes you on… it wont be long before you get noticed and nicknamed the yoyo dude… lol… happy throwing
edit: one more thing to mention… sizes… generally a larger size enjoys more stability and momentum, but dont sacrifice comfort… any midsize yoyo (midsize= a yoyo with a diameter about 2 inches… size points occer about every quarter inch so 2.25 or a yoyo with a close diameter is considered full size)