Learn Russian: Я хочу налить вам чашку чаю.

This is the partitive genitive, sometimes called the second genitive. When talking about a certain quantity or a part of an uncountable masculine noun (like a cup of tea, a piece of cheese), you will often see this -у or -ю ending. Saying чашку чая is also grammatically correct, but чашку чаю is a very common and natural-sounding way to express this in Russian.

Why do we use the dative вам here instead of saying для вас (for you)?

In Russian, the person who receives or benefits from an action usually takes the dative case directly. Just like in English where you can say "pour you a cup" instead of "pour a cup for you", Russian prefers the simple dative вам for everyday interactions like this instead of using a preposition.

Why use the perfective verb налить (to pour) here instead of the imperfective наливать?

You are talking about a single, specific action with a clear result: getting one cup of tea poured. The perfective налить focuses on completing that exact result, whereas the imperfective наливать would be used for the ongoing process of pouring or a repeated habit.