(covers information from several alternate timelines)
This page contains information regarding Star Trek: Picard, and thus may contain spoilers.
Apollo 11 was an American Apollo Program spacecraft that was in service with NASA in the mid-20th century. This craft carried the astronauts Neil Armstrong, Buzz Aldrin, and Michael Collins to Luna. Apollo 11 was launched atop a Saturn V rocket from Cape Kennedy. The original launch time was set for 6 am Eastern Standard Time on a Wednesday. (TOS: "Tomorrow is Yesterday"; PIC: "Two of One")
The lunar landing took place on July 20, 1969, at Tranquility Base. This mission was the first in the Apollo Program to successfully land Humans on Luna. The following day, Armstrong and Aldren became the first Humans to walk on Luna's surface. (TOS: "Tomorrow is Yesterday"; PIC: "Two of One")
The Apollo 11 lunar landing was shown on TV, a broadcast that the "whole world" had watched. Shannon O'Donnel was eleven years old at the time, and later recalled watching the landing on TV, and would also have dreams recalling Armstrong's famous words: "That's one small step for [a] man, one giant leap for mankind." In 2000, O'Donnell shared her moon landing experience with Jason Janeway, who himself had seen it once in science class. (VOY: "11:59")
When James T. Kirk argued in favor of having Sargon and his companions take over the bodies of the crew, in 2268, and emphasized the benefits that it might possibly have on mankind, he explained, "they used to say if man could fly, he'd have wings. But he did fly. He discovered he had to. Do you wish that the first Apollo mission hadn't reached the moon or that we hadn't gone on to Mars or the nearest star?" (TOS: "Return to Tomorrow")
A certificate was awarded to Cadet Harry Kim in 2369 while he was still at Starfleet Academy, related to the Apollo 11 quadricentennial. When temporarily given command of the "Nightingale", he felt this award was important enough to be one of the few things he brought along for his ready room, including his saxophone and his Academy diploma. (VOY: "Nightingale")
The three launch components of this mission were the command/service module Columbia (NSSDC ID: 1969-059a), the third stage of the Saturn V rocket (NSSDC ID: 1969-059b), and the lunar module Eagle (NSSDC ID: 1969-059c). The command/service module's name Columbia was later reused for a space shuttle and several civilian and Starfleet starships in the Federation. These include the Columbia NX-02, SS Columbia, and the USS Columbia. Likewise, the Federation starship USS Eagle was named in honor of the lunar module Eagle.
The Apollo 11 mission features heavily in the opening credits for Star Trek: Enterprise, namely with footage of Neil Armstrong and Buzz Aldrin walking to the launchpad, the Apollo Lunar Module landing, and Aldrin leaving a footprint on the moon.