Civil cases arising under federal law are generally filed in federal court: copyright cases, federal employment discrimination cases, cases against government agencies alleging violations of the U.S. Constitution, and more. But civil cases arising under state law can also be filed in federal court, if they involve more than $75K and the parties are all citizens of different states (known as "complete diversity of citizenship"). And if a plaintiff files such a case in state court, the defendant can remove the case to federal court, again if both the $75K requirement and the diversity requirement are satisfied. (This is of course an oversimplification.)
Now what happens if one of the parties is a business organization? Corporations are generally viewed as citizens of (1) the state in which they were incorporated, and (2) the state in which they have their principal place of business. But LLC's (limited liability companies) are viewed as citizens of all the states in which their members are citizens. That makes the complete diversity requirement harder to satisfy.
It also makes it important to know where an LLC's members—i.e., co-owners—are citizens. And if the members are LLC's, it's important to know the citizenship of their members. The result is pretty complicated.
