Glasses are tall and don't have handles. They can be made of glass or plastic. Стаканы are mostly used for drnking cold beverages like water or juice. Note that a wine glass (those with stems) are not called "стакан" - it is "бокал".
Cups usually have smaller heights and have handles. They are used for drinking hot beverages like tea or coffee.
There is also a word for "mug": кружка. But we often call mugs "чашки", too.
Hi, I hope I don't sound pedantic, but cups do not usually have handles, at least not where I come from. For me, "cup" and "glass" are interchangeable, unless the thing is actually made out of glass, then it should be called a glass. Or if it's made out of paper, then it's a cup. When it has a handle, it becomes a mug. Ahh, but wait, I see why you thought cups are small with handles--teacups! But teacups are the only case, the rest are mugs (coffee mug, beer mug, can't think of any more right now...). I just realized it could be quite confusing for non-native speakers:)
In Russia at least, glasses made of glass are often used for drinking hot beverages such as tea and coffee as well as cold beverages.
And in English, I suppose it's regional. I wouldn't ordinarily translate стакан as cup. Even though you might prefer to give your child a small, plastic cup of juice that could also be called a glass in English, that's not the most likely translation here.
сок is the nominative form and сока is the genitive form. Same word used in different situations.