For example, what role does Mark Cuban have as the owner of the Dallas Mavericks? Is it the same across all sports?
Owners of sports teams are just that -- they own the team. The best way to think about a sports team is as a business, because that is exactly what it is. From your example Mark Cuban can do whatever he wants. He can make player decisions like Jerry Jones of the Dallas Cowboys or be completely hands off. It's up to him. Ultimately it's a business and the team owner is trying to make the most profits possible.
Each North America major league team is managed differently, and unless you're an owner or a high ranking executive it's hard to tell what owners really do. For that reason the answer to this question is bound to be opinion based. This is my understanding of what the owners do.
Most major league sports owners purchase their teams because of the love of the game and sports in general, or as an investment (or a mixture of both). NBA is no different than other leagues when it comes to that. An example of how good of an investment owning an NBA team can be is the LA Clippers under Donald Sterling. He purchased the team just for $12.5 million dollars in 1981. He was more or less forced to sell the team in 2014 due to racist comments, but managed to sell the team for the wobbling amount of $2 billion dollars!
The role of an owner varies from team to team. Some owners are there purely for an investment, but others are more involved in running the team and the business. An example of a hands on owner is Jerry Jones the owner of Dallas Cowboys in the NFL. He bought the team in 1989, and not too long after he assumed the General Manger (GM) position. He has been both the owner and the GM of the Cowboys to this day. This means he has control over the business side of the team and also all football decisions, giving him complete power over all decision. You could argue there are no decisions that happen in that franchise without his approval.
But in general most owners (including those of the NBA) don't often make decision regarding how the team plays and functions. They usually watch the home games in the box, some travel with the team to see important road games or playoffs games. When the owners do make decisions for the team it is for the more important and long lasting ones. Their role when it comes to handling the team is to participate in the process of hiring new head coaches, presidents of sports operations, and sometimes players, as well as setting the roadmap for the teams future. Jim Irsay owner of the Indianapolis Colts is an example for an owner which is involved in the hiring and firing process of the coaches, GMs, and even the players.
To get back to your specification, Mark Cuban is more of a hands on owner. He's the type of owner that can talk about the players and their skills since he has a very good understanding of their performance as he can be found courtside of Mavericks' home games. He is involved in a number of different businesses and doesn't have the time or the deep knowledge of the game, hence he isn't as involved as someone like Jerry Jones.
Also the role of an owner can diminish if he or she is not the sole owner as they either don't have enough power to implement their decisions or have to get number of other people onboard. If you look at the list of NBA owners you'll see some teams are run as a family business like the LA Lakers (currently owned by the late Jerry Buss's family), and some other are owned by a number of different people like the Philadelphia 76ers.