As is explained in introduction to verbs and indirect objects section, a verb can sometimes take an indirect object, such as I give the ball to Ági. Here, Ági is the indirect object. Another example: Áginak adtam ezt a könyvet I gave this book to Ági.

In Hungarian the indirect object is usually placed in the dative case. Note that it's not just "to" that gets put into the dative case, as in the above example, but also "for": Veszek ajándékot Áginak I'm buying a gift for Ági

Some examples of obvious indirect objects:

• Viszek Áginak virágot I'm taking flowers to Ági
• Veszek ajándékot Áginak I'm buying a gift for Ági
• Nagymama köt nekünk pulóvert Grandma is knitting jumpers for us
• Fizetnek nekem egy sört They're buying me a beer

Some examples of less obvious indirect objects, where a "to" or "for" might sound unnatural in English:

• a magyar lány tetszik nekem I like the Hungarian girl. Literally, "the Hungarian girl pleases me" or "The Hungarian girl is pleasing to me".
• a magyar lányok tetszenek nekem I like Hungarian girls. Plural of the above. Literally, "Hungarian girls please me" or "Hungarian girls are pleasing to me".
• segítesz apunak? do you help (to) Dad?
• telefonálok Áginak I telephone (to) Ági
• örülök annak I'm glad about that (for that)
• nekem fáj a fejem my head hurts (to me). I have a headache.
• minden nap köszönök a postásnak every day I greet (to) the doorman.

Note, the dative is identical in form to the attributive/possessive case, but they perform different grammatical functions. They are different!

An interesting use of the dative comes when stating how someone is called (not what someone's name is, but what they are called). It's a little confusing because often the direct object is omitted (and the subject too).

When we say they call her Ági, the direct object is "her" and the indirect object is "Ági". This is translated as: Áginak hívják (they) call (her) Ági, in which the object (her) and the subject (they) have both been dropped. You could think of it as Ők Áginak hívják őt they call her Ági.

Note that we have used the definite conjugation here, because the object is "her". This requires the definite. If, on the other hand, the object was "me" or "you", we must use the indefinite conjugation.
This looks like: (engem) Danielnek hívnak they call me Daniel.

Both "Áginak" and "Danielnek" show the dative case, but note how we use a different conjugation depending on the (implied) object.

As a question, remember to put a suggested answer into the dative case too: hogy hívnak? Danielnek? What are you called? Daniel? .