What type of relay should I use to trigger an apartment door buzzer?
You need a relay with a normal open (NO) contact. You don't have to switch high currents, so a low power relay will do. Nevertheless the Arduino output will probably not be able to drive the relay directly (it wants too much current to activate), so you'll need a transistor to drive it, like this:  

 

Pick a relay with a coil voltage equal to the voltage you have available (the "+12V" at the top). The diode protects the transistor, and can be a 1N4148 (most common general purpose diode).
The relay contact of course goes parallel to the door-opener switch.  

This could be an appropriate relay.
edit: in this question Federico notes that there are different coil power versions of that relay. Unless you expect strong vibrations or shock choose the 150mW version.  

This relay may be even better: low coil voltage versions available, like 3V and 5V, and requires very low power.