When did they stop being popular?
Mega Man has always been cool!
On the surface, Mega Man is similar to any other platformer out there. You take your character through an obstacle course of spikes and enemies until you get to a boss at the end of the zone. You beat the boss and move on.
What made Mega Man unique, though, was the fact that every time you defeated a boss, you stole its weapon, too. On top of that, one of the other bosses would be vulnerable to another boss' weapon. So a big part of the challenge - and the fun - was learning which boss was vulnerable to what weapon, and the best order to tackle all the bosses. The different weapons also frequently allowed different ways to overcome some of the obstacles in the levels, so the more you had to choose from, the more imaginative you could get through each region. (For example, if a boss gave you a boomerang-type weapon, you could use it to snag power-ups and bring them to you from previously inaccessible portions of the zone.)
The Mega Man games also had a tendency to be surprisingly difficult, which to some gamers, adds to the appeal. But the variety of the weapons and the ingenuity of the weapons and enemies is what made the series so popular in the first place.