Translation:The director's bird is very colorful.
Basically, "az igazgató madara" is the simplified form of the complete form "az igazgatónak a madara".
The simplified form can be used when this structure is not "broken up". A simple adjective would not break it up ("az igazgató színes madara"), but a predicate would ("az igazgatónak színes a madara", or "az igazgatónak elrepül a madara").
When the structure is not this clean, for example, it is combined with some other structure, then you have to return to the full version, to keep the possessor-possessed structure apparent: "annak az igazgatónak a madara, amelyik .... ".
If you would use the genitive case in english or german, you have to construct a possessive relationship in hungarian. There are two ways. The easiest is to use nominative for the owner and possessive for the property. The other one uses dative for the owner.
Mother's book => az anya könyve OR az anyának a könyve.
Both are equal, but the dative form is more flexible to use. While the nominative form can only be used in that fixed sequence, you can switch with the dative form the sequence.
Az anyának a könyve => a könyve az anyának.
Or you can separate them by other words:
Az anyának kicsi a könyve. - Mother's book is small.
In some cases only the dativeform is useable:
Kinek a könyve ez? - Whose book is this?
Ennek a fiúnak a könyve. - It is the book of this boy.
The director has a bird - Az igazgatónak van egy madara.
The director has a very colorful bird. - Az igazgatónak egy nagyon színes madara van
The director's bird is very colorful. - Az igazgató madara nagyon színes.
The director's chair is very big - Az igazgató széke nagyon nagy.
Hope this help.
Is it just the course or is 'igazgató' used more commonly than 'főnök'? Generally I only think of movie and theatre directors and don't use the word much. I would use words like 'boss', 'manager' or 'principal' instead. You would get funny looks if your called these people directors.
It has to do with the general usage of the words. Főnök - boss is a relative description. You have the boss, but a company doesn't have a job description, "boss". Also bosses can be stacked: your boss can have a boss who has a boss themselves. On the contrary, igazgató - director is a fixed job. The company has a director, but you yourself don't.
"Principal" nad "manager" seem to be good translations for igazgató, too.