Mission Patches
Background
A tradition that grew up in the military was for a crew to design a patch that symbolizes their mission, and to use it to represent them and their mission. As the early astronaut corps was comprised largely of military pilots, it was natural that the same tradition should take hold in the space program.
Early Days
While it seems surprising now, earlier astronauts did not wear an American flag on their pressure suits -- just a name tag and the NASA emblem. However, like the "7" of Freedom 7, the idea caught on and has persisted to this day: all American crews since Gemini 4 have worn an American flag on their flight suits.
The next crew, Gemini 5's Gordon Cooper and Pete Conrad, wanted to find a way to personalize their flight, so Conrad hit on the idea of a patch -- such as Navy air squadrons have. With an ambitious 8-day mission planned, the longest to date, they designed a patch featuring a Conestoga wagon and the pioneering motto "8 Days or Bust". While NASA administrator Webb conceded to the crew their right to a mission patch, he decreed that the wording be deleted, lest a shortened mission be referred to as a "bust".
The tradition of a mission patch continues to the present time, with every Shuttle mission crew designing and wearing a patch usually signifying something unique about their flight.
The Popularization of Patches
While mission patches were initially an informal creation by the crew, they quickly became formalized, and were widely-used to symbolize each flight. Inevitably, space enthusiasts started collecting mission patches and, just as inevitably, entrepreneurs saw this trend as an opportunity to make money. The patch business was born. Patch collecting became so popular that entrepreneurs designed patches for all the flights that never had one. These patches are usually characterized by a stark lack of imagination and creative design.
While mission patches up to and including Apollo 1 were embroidered, following the tragic Apollo 1 fire, all flammable materials were banned from the spacecraft, and this included embroidered patches. Instead, beginning with Apollo 7 all crew patches were silk-screened onto non-flammable "Beta cloth". Embroidered versions of each patch were still used by astronauts and ground support personnel for non-flight use [Still, p. 168] - and purchased by collectors. Unfortunately, in the rush for profit, the creators of embroidered patches were seldom concerned with fidelity to the original design.
It's important to note that, as with their embroidered equivalents, when the spacesuits worn on a flight are returned to Houston, the flown patches are removed from the suits, given to the astronaut who wore the suit, and replaced with unflown (but otherwise identical) patches.