The first grenade launchers appeared back in the flintlock era and were bell-mouthed blunderbuss like weapons that fired the "flaming onion" hand grenades of the era. The grenades themselves weren't all that effective and the weapons themselves were short ranged, inaccurate and generated a lot of recoil for the results they gave. Lighting the fuse and balancing the time taken to fire with the need for the round to burst soon after impact were also a problem. Eventually people went off them and they disappeared into museums.
Then the grenade reappeared and people started looking for a way of getting it to travel further than it could be thrown by hand. First came the rifle grenade and then came the (informal) rifle dedicated to the job of grenade launching. Eventually people started thinking in terms of a purpose built weapon - and in the meantime several of the light mortars used in and before WW2 were called grenade launchers (including the Japanese "knee mortar", which probably deserves a special mention in the heredity of the modern grenade launcher).
In 1961 the US introduced the M79 - arguably the first modern grenade launcher - a break action design sort of like a big, single barrelled shotgun and everyone else seems to have followed on from there2.
These weapons don't actually fire hand grenades - they fire a stubby, bullet like thing which occasionally earns them the nickname 'spud guns'. That bullet like thing is a launcher grenade.
Launcher grenades come in pretty much any flavour you can get a grenade in - including anti armour, multiple projectile and flares as well as more conventional things like explosive and smoke. Some cutting edge designs include thermobaric rounds and miniature remote pilot vehicles fitted with cameras.
The launchers themselves vary from single shot copies of the M79, through revolver and pump action weapons to fully automatic, belt fed monsters. Some of the most popular designs are single shot weapons designed to mount below the barrel of another long arm - such as an assault rifle - so that an infantry section can gain additional firepower without sacrificing a rifle.