The Catholic Church gets its money from several sources. One is the collection in each mass. Sometimes they collect more than once, as, for example, when they want to do some improvements in the building or when the collection is meant for a special purpose (for example, to give money to Mother Teresa's nuns). There's no fixed amount and no real obligation to give. I speculate that this must be relatively pocket money, particularly because most Catholics live in poor countries (say Africa, Latin America), and the numbers of Catholics are dwindling in Latin America, though they are growing impressively (or so Catholics say) in Africa.
There's also people who give money to the church, for several reasons. This isn't compulsory either. You can donate for, say, the building of a new church, or the publishing of a number of books, whatever comes to mind. Even for the celebration of a patron saint.
Donations and collections from richer countries are significant, for example those coming from the United States, Canada, France, Ireland, Austria, Germany, Spain, Italy, Mexico, South Korea (yep).
There's also hospitals, clinics, orphanages, schools and the like that get support from richer Catholics, sometimes governments, other dioceses, et cetera. I don't think any of this money goes to the administrative structure of the church (the Vatican, the local dioceses or archdioceses). In some countries Catholic schools are also elite schools, and the rich send their children to study there, which means they pay tuition. Let's not generalize about all Catholic schools being for the rich, because I studied in several different schools and none was for the rich. They also have many universities around the world. Some of those have at least local reputation. And then think Georgetown, Louvain, and the not so prestigious but still meaningful Catholic universities around the world. The Jesuits have so many, and Jesuits do like money and power. The Opus Dei also has at least one university that I know of, in Navarre, Spain.
Because of the work of their founder with children, salesians have many elementary and high-schools and shops where the youth can learn practical skills, such as carpentry. To my knowledge, they do admit students from other religions, though sometimes they insist that they participate in prayers, which of course the faithful of other churches won't do.
I understand German Catholics pay a percentage of their money to their church. That must be why German bishops give money to dioceses in poorer countries.
Catholics have NGO's as well. Think Catholic Relief Services and Caritas, for example, which are active worldwide. I don't know where these NGO's get their money from, but I assume donors would be very careful not to give money to the Vatican.
Some churches also get money for masses. Say you want the priest to say the mass in memory of your loved one: though it is not mandatory that you give an amount of money, you can do it, as a "contribution". This very much depends on the parish priest (the top guy, because in some areas there's more than one). Sometimes you are even forbidden from donating this way, and sometimes the priest is more than eager to receive your money. I don't know how much this would amount to.
Then there's the impure ways to raise money, such as the church having a bank (a legitimate business) that is said to serve as a money-laundering institution (not so much). I don't know how many of you know that John Paul I is said to have been killed, for different reasons, and among the names mentioned as potential killers were Paul Marcinkus, an American bishop who was a financist, Roberto Calvi, an Italian banker and mason who was found hung in London, with several bricks in his coat's pockets, and Michele Sindona, who is said to have had ties to the Italian Maffia and died/was killed while in prison. The English author Bishop Marcinkus had invested the church's money in many concerns, a condom factory among them. They also had a lot of real estate. And you need to remember that for many centuries the church was a huge landowner in many countries. It owned a respectable percentage of the entire country of Mexico, for example. Source: David Yallop, "In God's Name". The murder was not proved but the financial operations are for real.
A good deal of the money has gone to legal settlements on cases of child abuse.