Unlike other slavic languages, Bulgarian does not generally have a case system. However, like English, Bulgarian preserved remnants of that system in the personal pronouns.
In English, when you say "I am", "I" is the nominative case of the 1st person singular personal pronoun, because it is the subject. When you say, "He told me", "me" is the accusative case of that same pronoun, because it is the direct object of a transitive verb ("he" is the subject). When you say, "He gave them to me", "to me" is a dative case, because it is the indirect object ("them", the direct object, is the accusative case of "they").
He gave them to me ----- ----- ----- Nominative Accusative Dative
|to me||на мен/мене||ми|
|to you||на теб/тебе||ти|
|to him/it||на него||му|
|to her||на нея||ѝ|
|to us||на нас||ни|
|to you||на вас||ви|
|to them||на тях||им|
- When used for politeness, вас/ви takes a capital letter
- мен and мене, and теб and тебе are interchangeable
In modern Bulgarian, the long forms of dative cases are replaced by the preposition "на" followed by the long form of the accusative case. Note how the short dative pronouns are the same as the short possessive pronouns.
If you are a bit confused about whether you should use the accusative or the dative case, remember that the long dative forms use the "на" preposition, so the short dative forms can only be used where you would have used "на" + accusative in the long form.
Long forms come after the verb:
"Аз питам теб" - I am asking you
"Аз говоря на теб" - I am talking to you
"Давам ябълката на тях" - I give the apple to them
As a rule, the short forms are placed in front of the verb. However, since they are not normally accentuated, they can't appear as the first word of the sentence and sometimes have to be moved after the verb:
"Аз те питам" or "Питам те" - I am asking you
"Аз ти говоря" - I am talking to you
"Давам им ябълката" - I give them the apple
Generally, the short forms are prefered to the long ones, unless the long form is required by a preposition such as "с", "на" or "за". Even though it's not incorrect to use the long forms when the short ones could be used instead, it can sound weird, especially when there are several pronouns in the same sentence.
As a side note, you may remember from the Possession lesson that the short possessive pronoun sometimes suspiciously appeared before the verb in some sentences:
"Той ми е братовчед" - He is my cousin
The possessive pronoun would normally appear after the noun but in this case, it appears before the verb because it is in fact a dative case: this sentence literally means "He is a cousin to me".