The short answer is a 50-year period in which Rome became embroiled in defensive border wars and internal instability.
When did it begin? It depends on who you ask. The general consensus is that it began with the death of Alexander Severus in 235 and ended with the ascension of Diocletian in 284. Alexander was killed by his troops, who proclaimed Maximinus Thrax emperor.
The Senate wasn't too happy about that, especially because Thrax wasn't exactly "Roman." He was probably Scythian or Gothic, and he had no real political connection to Rome. He hated the nobility, and when the Senate tried to replace him, there were some...setbacks involving four other people being proclaimed emperor and dying badly. Thrax died in a mutiny while he marched on Rome to reassert his authority.
From then out, we had a quick succession of emperors. Many were born outside Rome, were elected by the military, and only grudgingly confirmed by the Senate (if at all). Many spoke Latin with "barbarian" accents, and some never even set foot in Rome until it was time to cement their claim. When it comes to the guys between Gallienus and Diocletian, they all probably knew each other to some extent, which makes the infighting and backstabbing all the more saddening.
Me? I'd actually place the beginning with the death of Commodus and the civil war that followed in 193. We had five proclaimed emperors that year, one who was elected by the Senate and killed by the Praetorian guard, one who actually won the title in an auction, and three who'd been proclaimed by the legions in the field. The winner wasn't popular with the Senate, and he didn't like them much either.
In any case, it's a complex and sad tale. We had a few decent guys (Gordian III, Alexander Severus, and Numerian), who would have made good emperors in peacetime but were doomed in times of strife. We had an emperor named Philip, who was sympathetic to Christianity. We had guys who had a real grudge against Christianity, like Decius and Valerian. Aurelian's own inflexibility would lead him to defeat two breakaway empires, but would lead to his death. Carus might have died of a lightning strike. Elagabalus pronounced himself the voice of a god. Gallienus had somewhere between 19 and 30 usurpers to the title during his reign. As Agrippa mentioned, Valerian was taken hostage by Shapur and never saw the empire again (that wouldn't happen again until 1071).
Wikipedia has a decent synopsis here. It helps to have this list open in a separate tab.