The simple answer is: "it's very complicated". Individual churches, dioceses, national councils of bishops, religious orders, houses of those religious orders, the dicasteries of the Roman Curia, institutions like schools and hospitals: they are all to some extent independent and to some extent related. The Catholic Church is not a simple organisation, and it doesn't have a simple financial structure.
In particular, it is complicated because the exact pattern is not replicated in every country. The national Episcopal Conferences have some degree of autonomy in setting a pattern which will apply in each country, according to what is appropriate. Moreover, the Vatican finances are both complex and opaque. Some income comes directly from donations from the faithful (a system known as Peter's Pence), while other income comes from the surplus from the Istituto per le Opere di Religione, the Vatican Bank.
There isn't a simple organisational structure, with the Vatican at the top and the parishes at the bottom and a simple flow of money up and down the chain. This is because the basic structure of the Catholic Church is based around is that of the particular church. This is generally (though not always) a diocese. It is local (that is to say, defined by geography) and headed by a bishop. (Code of Canon Law, Canon 368) If you want some of the theology behind this, the document Communionis Notio is relevant.
The diocese are institutionally independent of one another, though they are grouped in the national/regional Episcopal Conferences which have certain legislative powers in their region, and they are all subject to the Canon Law of the Catholic Church. Each diocese is divided into parishes. The priests in each parish are essentially "deputies" for the bishop.
The financial relationship that we might consider best, therefore, is between the diocese and the parish. (The financial organisation of the Vatican would take years to analyse fully!) The parish-diocese relationship is the most common financial relationship. It leaves a whole lot out, as you see from my opening paragraph. So let's look at one such example. I'm going to look at the Roman Catholic Diocese of Westminster and it's 2011 financial statement, for the simple reason that I could find it online.
The parishes receive income of various types (p19):
- collections, donations and legacies (by far the largest)
- parish activities
- investment income
- trading income
- disposal of assets
This is spent in various ways (p21). The largest are non-clergy salaries and parish assessments, which are in effect a diocesan tax: each parish provides a proportion of their income to the diocese. This proportion varies among Episcopal Conferences and possibly among dioceses: I haven't found any clear information.
In each parish, the parish priest is primarily in charge of finances:
In all juridic affairs the pastor represents the parish according to the norm of law. He is to take care that the goods of the parish are administered according to the norm of cann. 1281-1288. (Canon 532)
On top of this, there is to be a finance council in each parish:
In each parish there is to be a finance council which is governed, in addition to universal law, by norms issued by the diocesan bishop and in which the Christian faithful, selected according to these same norms, are to assist the pastor in the administration of the goods of the parish, without prejudice to the prescript of can. 532. (Canon 537)
So many decisions are taken on a local basis, by the priest with the advice of the finance council.
At a diocesan level, most income comes from the parish assessments mentioned above (p20 of the Diocese of Westminster report). In the diocese, there is again a finance council to manage finances:
In every diocese a Finance council is to be established, offer which the diocesan bishop himself or his delegate presides and which consists of at least three members of the Christian faithful truly expert in Financial affairs and civil law, outstanding in integrity, and appointed by the bishop. (Canon 492)
The function of this council is to provide "a budget of the income and expenditures which are foreseen for the entire governance of the diocese in the coming year".
So a short summary.
Most income comes to individual parishes. The parish priest is in charge of this, with the assistance of a finance council. A proportion of income goes from the parish to the diocese, where a finance council appointed by and presided over by the bishop is in charge of decisions concerning the governance of the diocese.
This is obviously not the whole picture. It covers only the parishes and diocese. These are the most significant aspect of the Church, but a lot more could be said about the various other aspects of the Church on local, national, regional and international levels. Moreover, to some extent these relationships vary significantly across the world.