I don't know how the phone system works in Poland, and besides that's a question for a telephony user group or something, not a linguistics question. So I will address the linguistic point.
Many languages other than English have a case system for marking the grammatical roles of words like nouns. In most instances this also applies to proper nouns like personal names. Case-marking in Polish takes the form of suffixes that are appended to the base form of the word. For the masculine name Dawid, the forms are:
Nominative Dawid = David as a subject
Accusative Dawida = David as a direct object, to David
Dative Dawidowi = David as an indirect object, to/for David
Genitive Dawida = David's, of David
Instrumental Dawidem = with/by David
Locative Dawidzie = at/on/to David
Vocative Dawidzie = O David!
Apart from that, there exist prepositions, as English to, from etc. The verb (za)dzwonić 'call' takes the preposition do 'to' which requires the genitive, so the whole phrase is (za)dzwoń do Dawida 'call to David-Gen.'.