This Church is many things - institution, community of disciples, servant to the world, herald of the Good News, mystical Body of Christ, People of God etc. Democracy is about political organisation, so this question involves it primarily as institution, and a little as community.
The Church is 2000 years old. How it is meant to be now is influenced by how Jesus meant it to be, and human understood of our relationship with God today.
How Jesus intended things is not obvious - scripture and early tradition do not have direct instructions about how to organise ‘the church'. But it does have the principles Jesus taught, eg love of neighbour, concern for the poor. However historical evidence provides views of how early Christian communities operated, and viewing scripture in light of its political and cultural context provides insights into how the disciples interpreted Jesus intentions for the institutional structures they needed.
The Roman empire was powerful, and authoritarian. Governors ruled of local areas, according to the will of the Emperor. Slaves had no legal rights. Against this, an organisation which took seriously complaints of powerless members eg widows who were missed in the daily bread distribution, appears almost revolutionary.
History indicates the organisational forms which must have been known to the disciples as they organised the first churches. The Greek ideal democratic state involved land-owning adult males in decisions on a one-man, one vote basis. Historical sources show that some of the local early churches used voting methods to select their leaders.
Thus, while the early Church was not democratic in either the classical Greek model (widows voices as well as those of men were heard, and the Patriarch of Rome who became acknowledged as the ultimate source of 'truth' was not elected), it was certainly strongly influenced by democratic practices.
The Church through the periods between the early centuries and today was clearly not democratic, whether or not it was meant to be. Once Constantine converted, the church was aligned to government. Often it wasn’t possible to distinguish between the two. Democracy did not feature in secular politics until the French Revolution. Pius IX, in Mirari Vos, made it clear that democracy was not favoured by the Church. Bishops were appointed in various ways (by the monarchy, the nobility, the Pope) - but certainly not by popular election.
Today's Church is clearly not democratic. Bishops are appointed by the Pope, who is elected by officials selected by the previous pope. The Catechism of the Catholic Church has a section on 'Christ's Faithful-Hierarchy, Laity, Consecrated Life" which makes it clear that Bishops solely govern their diocese. Selected laity and religious are consulted, but it is not always clear if their views influence decisions. Lumen Gentium discusses the role of bishops at length, describing them as "placed in charge of particular churches" (para 23). It also states that the apostles appointed their successors and that Bishops today are result of apostolic tradition (para 20). Canon Law makes it clear that priests have ultimate pastoral authority over their parishes, with the power to over-rule pastoral councils (democratic or otherwise).
Whether today's Church is meant to be democratic is more complex. Secular politics do not point towards democracy as particularly efficient (though neither do they suggest a better method). Democracy requires educated voters, and while the baptised today are better educated than 100 years ago, they do not live and breathe scripture as the Jewish people did, for example. There are certain democratic aspects that are clearly supported by Catholic social teaching. The ‘common good’ says that the needs of all people must be considered - this is democratic in a sense. The same social teaching, though, points out negative aspects of the 'tyranny of the majority'.
Overall, I must conclude that the church is not democratic, in either the original sense of the word, or in today's broader adult-franchise sense. But the church clearly is meant to embrace certain aspects of democratic tradition, and has done so in a number of occasions.
Lumen Gentiuum - the Dogmatic Constitution on the Church. Documents of Vatican II, 1964.
Catechism of the Catholic Church 1994. CEPAC edition.
A Concise History of the Catholic Church. Bokentoter, T. 1990.Doubleday, New York