Shura (Arabic: شورى shūrā) is an Arabic word for "consultation". The Quran and the Prophet Muhammad encourage Muslims to decide their affairs in consultation with those who will be affected by that decision. Shura is mentioned as a praiseworthy activity often used in organizing the affairs of a mosque, Islamic organizations, and is a common term involved in naming parliaments.
Shura, as done during early Islamic history, is not the same thing as modern voting, a modern parliament, or modern democracy. Modern democracy developed recently in history, and some Muslims back-project it and claim it has roots in Islam by saying it is like shura. However, modern democracy is different from shura, and this is not an accurate analogy; rather, it is a form of apologetics.
In any case, the Shii belief is that the succession to the Prophet (S) was appointed by Allah and people did not have the right to choose it - neither by shura nor by a one-person-one-vote system - just as people do not have the right to choose how many times a day they say formal prayers, or what month to fast in.
However, Allah has not appointed specific individuals to be our political leaders presently, so we use other systems of governance and selection of leaders.
Furthermore, when you read the history of what actually happened during the selection of the first three caliphs, you see that some things happened that were questionable or politically motivated for an agenda, and it was not that everyone got together and simply discussed with sincerity and equality who should be the next leader. The scenario was different for each caliph. When you read what happened, you see it is nothing like voting today where every person has a single vote and the majority wins. You can look into that in history books.