Resolved questions
How is the dative case used in Russian and how is it formed?

Dative case has such questions to the object, as --- to give wich ? to give what?

+ Кроме того, дательный падеж используется после предлогов «к», «по», «благодаря»:
Я иду к другу.
Мы гуляем по Невскому проспекту.
Благодаря твоей помощи я сделал работу вовремя.

Dative is a Russian grammar case that expresses action is addressed or referred to the object you are speaking about. Here I'll try to explain the order of forming it.

Mother gave а bottle of water to (WHOM?) her child.
Мама дала бутылку воды (КОМУ?) своему ребенку.

Here the word "Ребенок" was used in a dative case "РебенКУ", because the action that mother did ("GAVE") was addressed to some object ("CHILD").
In dictionaries all the words go in Nominative. So, to make a dative form of it, you should learn the rules of changing the word's ending according to its gender and form (singular or plural). It's a difficult thing to explain in two words.
Hope my answer was helpful.