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This is my understanding of the Supreme Court ruling on this matter (IANAL, etc.).
Basically, the travel ban is reinstated only for individuals who do not "have a credible claim of a bona fide relationship with a person or entity in the United States" (p. 12). The Court's reasoning being that, if such a relationship is present, the harm caused to the U.S. person or entity in question by not allowing the foreign individual to enter outweighs the Government's interest in maintaining national security.
What constitutes "a bona fide relationship"? If the relationship is with a U.S. individual, "a close familial relationship is required." If it is with a U.S. entity, it "must be formal, documented, and formed in the ordinary course, rather than for the purpose of evading" the ban; the Court further explains:
The students from the designated countries who have been admitted to the University of Hawaii have such a relationship with an American entity. So too would a worker who accepted an offer of employment from an American company or a lecturer invited to address an American audience. Not so someone who enters into a relationship simply to avoid [the ban]: For example, a nonprofit group devoted to immigration issues may not contact foreign nationals from the designated countries, add them to client lists, and then secure their entry by claiming injury from their exclusion.
State Department guidelines issued to U.S. embassies and consulates and reported by The New York Times make the further two clarifications:
Note that even if you do not have such a relationship, this does not necessarily mean that you can't enter the U.S.; only that the Executive Order applies to you as written, including its numerous exceptions (so if one of those exceptions applies to you, you can enter).
For pending visa applications, if you are found to be ineligible for entry to the U.S. in light of the ruling (i.e., if you do not have a relationship with a U.S. person of entity and none of the exceptions provided in the Executive Order applies to you), you become ineligible for a visa as well. Otherwise, your application is unaffected by the ruling.