Henchmen and followers are concept that seems to be most common in various versions of Dungeons & Dragons and the many other games based on its rules (Star Wars Saga, Conan d20, d20 modern, and so on). It's rarely used in RPGs, even in games that have specific rules for it, but is quite common in fiction.

Henchman: A character with most of the abilities of a hero, who can provide direct assistance in battle or help out in other crucial ways, but lacks the ability to completely take over the role of one of the heroes.
Follower: A character who lacks any special abilities of a hero and is not capable of driving the story forward in a meaningful way. Their role is mostly to provide support in the background the heroes can't be bothered with and may be able to hold off faceless enemy minions as a group.

Depending on how many followers there are, most of them at least have a name and a trace of personalty. The hero they follow is assumed to know them personally, even if i never comes up explicitly. They are also loyal and don't just run away when things get too dangerous or someone makes a better offer. When a heroic commander or warlord gets deserted by his armies, the followers are the last people still with him.
Very often such characters are seen as the crews of ships. Like the crew of the Black Pearl in Pirates of the Caribbean, or the minor characters on the Normandy in Mass Effect. Conan also once shares command over a pirate ship in Queen of the Black Coast, and frequently is the leader of groups of raiders. And then there's of course Aragorn and his Dunedain or Robbin Hood and his Merry Men.

Henchmen are quite common as sidekicks for superheroes. Or they kind of take the role of the heros manager, who takes care of the base when the hero is doing adventuring. There's lots of them all over Star Wars. R2-D2 being the henchman to Luke, and though he does a great job in a fight, Chewbacca also is one for Han Solo. Most of his role is taking care of the ship and covering Hans back, but he doesn't really do anything that drives the plot forward. Lando and Jabba also have their assistants, and given the relatively big role, and occasionally important role he plays,even C-3PO could be counted as Leias henchman instead of one follower among many. Then of course Batman has his Alfred and Jack Sparrow got Mr. Gibbs, who takes most of the speaking parts of the crew, and again takes care of the ship when the heroes again. And Mass Effect has Joker, who is the pilot of the ship and probably has more than twice as many lines as the whole rest of the crew combined. But the only thing he ever does is flying the ship (which is a major part in the story at many points), and complaining about everything.

I think henchmen and followers could possibly be great addition to a campaign, without players ending up playing two characters. Does anyone have experience with them or ideas how to get them introduced and involved in a campaign?
They probably work best in campaigns where the PCs are not just dungeon crawling wandering adventurers. For henchmen and followers to have something to do, things need to happen in places where the PCs are not currently, too. And there need to be things to do, that are not fighting powerful creatures.