Before the widespread adoption of radio, carrier pigeons were frequently used on the battlefield as a means for a mobile force to communicate with a stationary headquarters. They saw considerable service in WWI and were still in use in WWII. But their use far predates the 20th century. Cyrus the Great used carrier pigeons to communicate with various parts of his empire. During the Golden Age of Greece pigeons were used to carrier news to other cities of the outcome of the Olympic Games. Anacreon wrote an Ode concerning a carrier-pigeon. In Ancient Rome, within many texts, there are references to pigeons being used to send messages by Julius Caesar, though I don’t recall that big Juli mentioned pigeons in Caesar’s Gallic Wars. In 12th century Baghdad and Egypt there was an organized pigeon messenger service. Genghis Khan also used them. In 1436 the Spanish traveler Pedro Tafur mentioned the use of carrier pigeons in the Ottoman Empire. In Europe, the Republic of Genoa had watchtowers equipped with carrier pigeons.
The domestic pigeon (Columba livia domestica) is a pigeon that was derived from the rock pigeon. The rock pigeon is the world's oldest domesticated bird. Mesopotamian cuneiform tablets mention the domestication of pigeons more than 5,000 years ago, as do Egyptian hieroglyphics. Research suggests that domestication of pigeons occurred as early as 10,000 years ago. Initially pigeons were bread for meat which in English is generally known as squab. I have eaten pigeon--didn't fancy it. It was by far the worst meal I ate in the UK.
A good racing pigeon can average 60 mph through the air for hours at a time, and fast racing pigeons have been clocked at 110 MPH. As far as distance, pigeons have been known to travel 1,000 miles or more.
In fiction, the 1935 version of The Three Musketeers (which I happened to watch recently) uses messenger pigeons more than once as a means of fast communication. The villain of the film is Rochefort and we see a scene where dozens of pigeons are sent out to alert his agents to so that they can ambush D'Artagnan and the Musketeers on their way to England to get the Queen's diamonds. Later we see Milady de Winter send a message back to Paris by pigeon. Recently carrier pigeons were used in my game as a means of fast communication between Cardinal Richelieu and his agents.
As far as roleplaying, carrier pigeons are not something I've seen in RPG materials. The closest to carrier pigeons is from some Runequest setting materials that mention Yelmalio cultists using hawks to carry messages. Of course in many RPG settings magic provides an even faster method of communication (though one that requires mages) and high power D&D may use hippogriff or pegasi messengers.