Short answers from a non-Russian:
The partitive that resembles the masculine dative (i.e., with -у/-ю) is extremely rare now. Historically, there were many dozens of masculine nouns that had this special form, but in in modern speech, чаю and сахару are probably the only examples you're likely to hear. Otherwise, the partitive will look just like the genitive, regardless of the noun's gender (and in the case of "tea" and "sugar", it's perfectly acceptable to say чая сахара, the normal genitive forms).
Basically, the partitive expresses the idea of "a small quantity" (i.e., "a bit of; a little; some; a morsel of") without needing to use a quantifying word like немного ("a little") or кусок ("a piece") or ломтик ("a slice") before the noun. In other words:
Я выпил воды (partitive genitive) means the same thing as Я выпил немного воды, namely "I drank a little water" or "I drank some water".
On the other hand, Я выпил воду (normal accusative) is closer to "I drank the water" (it might have been just a little water, or might have been quite a lot).
Similarly, Она съела сыра (partitive genitive) means "She ate a little bit of cheese", and you could express more or less the same idea by saying Она съела тонкий ломтик сыра, "She ate a thin slice of cheese." But Она съела сыр (accusative) can suggest a large amount of cheese.
One important thing about the partitive is that it's used with mass nouns (i.e., "water, wine, cheese, flour, chocolate, sour cream...") that can be divided into portions but that don't usually have a specific number such as "one" or "five" or "thirteen" in front of them, but not with singular count nouns (i.e., "an apple" or "an egg"). However, plural count nouns can be used in the partitive -- for instance, яйцо is "an egg," and its genitive plural яиц can be used partitively to mean "a few eggs".
Hope this helps!