The short answer in an injury that is actual or imminent, particlarized to the plaintiff, caused by the defendant, that is within the Court’s power to redress.
The first requirement of federal standing, as stated by zev, Whack-a-mole, and Poly and others on preview, is injury, or more specifically injury in fact, an injury that is actual or imminent, not merely hypothetical or conjectural. “Injury in fact” is a constitutional requirement, and Congress can’t do away with it by legislating, say, “if so-and-so violates this law and illegally dumps toxic waste, any U.S. citizen can sue him.” The case or controversy requirement mandates that the litigating citizen still show how he is actually injured by the dumping.
The next constitutional limitation on standing is that your injury must actually be caused by the defendant. This is not always as simple as it sounds. For example, in Allen v. Wright, parents of black schoolchildren claimed that their children were not getting a good education because the IRS was not enforcing laws that would deny tax-free status to schools that racially discriminated. The Court said that the link between the governement’s conduct and the injury complained was too distant.
The last constitutional requirement is redressability; the Court must have the power to do something to eliminate the harm to the plaintiff. For example, I can’t challenge tax breaks given to hospitals that won’t provide me indigent medical care unless I can show that I would be receiving it if the tax breaks were not provided. The Court doesn’t have the power to take action that will allieviate my injury, so I have no standing. One sort of quasi-exception to this rule is when the injury is of a sort that is “capable of repitition but evading review.” For example, if you are challenging your state’s prohibition on abortion, you will likely have already had your child by the time the case gets to the U.S. Supreme Court. In a rare circumstance like this that could repeat itself over and over, effectively denying any litigant from ever bringing suit, the Court will find standing, even though in this one instance redressability is arguably moot.
In addition to constitutional limitations on standing, there are also prudential limitations created by the judiciary. These are cases where they could hear the case unless (1) instructed by Congress or (2) there’s a really good reason to. One prudential limitation is against the “generalized grievance,” especially as it relates to taxpayers. You must have some particularized injury that sets you apart from the general populace, and not just some generalized complaint that any citizen would have that the government is acting unconstitutionally. For example, you would have standing to litigate the amount of taxes you personally owe, but not standing to litigate that Congress is unconstitutionally spending your taxes in Iraq. Taxpayers have virtually only one instance in which they can challenge the unconstitutional use of their taxes – when they are challenging the aid to religious organizations under the Establishment Clause.
Another prudential limitation is when a third party is asserting the rights of others. This will be allowed when a special relationship exists between the party and the non-party would have difficulty asserting his/her/its own rights. For example, the NAACP was allowed to sue on behalf of its individual members in a suit challenging a state law that required disclosure of its membership. If the members had sued individually, they would have revealed their identities, thus making the lawsuit moot.
Which brings up organizational standing. Generally, an organization can sue when it has suffered harm as an organization. In some instances, however, the organization may also sue on behalf of its individual members if both have been
injured in a manner related to the purpose of the organization, so long as the individual participation of the members in the lawsuit is not required.