How did client/patron relationships work in ancient Rome?

The patron client relationship is extremely complex, there was no 1 size fits all kind of agreement. The easiest way to explain this relationship in layman terms is through the movie "The Godfather". The scene where people are going to the godfather at his Daughter's wedding to ask for a favor which he will grant, in return for a favor from them sometime down the road. This is essentially how the patron-client relationship worked. The patron would receive clients in the morning at his home and give them various favors such as money, food, etc... These kinds of clients usually would be expected to return the favor by voting for their patron's candidate in the upcoming election. That is the very basic level of everyday client patron relationships. However, as I mentioned this is a very complex subject and relationship. For instance defending someone in court was a way to either repay a favor or to earn favors. Whenever someone passed away, their clients passed to their legal heir(s). This allotment could be specified in their will. So for example when Julius Caesar was murdered, the reason why 18 (just shy of his 19th birthday) Gaius Octavius suddenly became a political force was because he inherited Caesar's clients (among other prominent factors, such as his name). That was a big deal because the dictator was known for his clemency. Think of all of the prominent men who faced Caesar in the civil war and were spared their life by him. They became Caesar's clients in a huge way, since they literally owed him their life. This was part of Octavian's inheritance. Freedmen owed their freedom to their former masters, so they and their families would usually end up as clients of their former owners for generations. Some other big ways this relationship worked would be in loans, prosecuting someone in the courts on behalf of a non-citizen, granting someone citizenship, etc...