Morale patches are patches troops wear on their uniforms designed to be a funny inside joke, applicable only to their unit or military career field. They are usually worn during deployments, but the wear of morale patches is at the discretion of the unit's commander.
The patches often (not always) make fun of a depressing, boring or otherwise specific part of the job.
These patches have been around since the military began to wear patches. They are collected and traded by people, both military and civilians, who come across them. Some are more popular than others, but they are usually a lot of fun.
The "Morale Stops Here" patch is pretty popular and is actually repeated by units the world over. It's really funny the first time you see it.
This is an old one, a throwback to the Air Force's Strategic Air Command days. "To forgive is not SAC policy" is widely attributed to famed SAC commander Curtis LeMay.
Having the Kool-Aid Man as your unofficial mascot is funny enough, but making his hand the lightning-shooting gauntlet in the old SAC emblem is clever.
The JSTARS (or Joint Surveillance and Target Attack Radar System) have a descriptive patch here - as they operate out of trailers at Al-Udeid Air Base, Qatar (in the military, being deployed here is also known as "doing the Deid").
This is a U.S. Navy patch from Vietnam. The "yacht" is a junk - a historically widespread type of ship used in China and around Southeast Asia. The Tonkin Gulf is where the Vietnam War (or more specifically, the U.S. involvement in it) really ignited.
More from Vietnam. By the end of the 1960's, the rift between those who served in Vietnam and the perception of the war back home hit its peak.
As the Cold War intensified and the threat of nuclear war seemed more and more unavoidable, the young enlisted and officers whose role in the annihilation of Earth's population probably felt more than a little stressed.
The tradition continued, well into Desert Storm. If you have morale patches that make others laugh or are highly prized, please post in the comments.