What are the bosses' weaknesses in Mega Man 2?
Okay, so unlike the other games, there's no strict cycle in the game.
The question that people have linked in comments describes the particular order you probably will want to go, but here's a readout on the actual weaknesses. We'll take it a step further than boss ordering by actually including the Fortress Bosses in there.
Metal Man - Instant kill from Metal Blade. Since you can't get that on the first run, the options are Quick Boomerang or Atomic Fire; Metal Man actually takes the least damage from Atomic Fire from all the eight bosses that do take damage, but its fully charged shot is as powerful as the Quick Boomerang.
Bubble Man - Primary weakness is Metal Blade. You can settle for half its power with Crash Bomb or Quick Boomerang.
Heat Man - Primary weakness is Bubble Lead. Air Shooter can work as an alternative, though it isn't nearly as strong.
Wood Man - Instant kill from Atomic Fire when fully charged. Secondary weakness is Air Shooter humorously, but most people will instead opt for Metal Blade which is still high damage despite being half as powerful.
Air Man - Primary weakness is Leaf Shield. Atomic Fire is the only other real option, but he does take increased damage from your Mega Buster.
Crash Man - Primary weakness is Air Shooter. Doesn't really have a secondary weakness.
Flash Man - Primary weakness is Metal Blade. Crash Bomb is almost as powerful, but not quite.
Quick Man - Primary weakness is Time Stopper. However, it uses up the whole gauge just to do half his health; Crash Bomb operates the standard weakness game, although Atomic Fire does impressive damage when fully charged. Like Air Man, Quick Man also takes increased damage from the Mega Buster.
Dragon - Doesn't actually have a strict weakness. Quick Boomerang is often suggested due to its arc. A charged Atomic Fire will do impressive damage, though.
Wall Robots - Their technical weakness is Bubble Lead, which instantly destroys them. Most people instead opt for Metal Blade, since it can be aimed despite needing two shots.
Guts Dozer - Weakness is Quick Boomerang, but a charged Atomic Fire also puts a large dent.
Sentry - Crash Bomb is the only weapon that works at all.
Wily Ship First Phase - Atomic Fire fully charged should do impressive damage (maybe even destroy him, if my memory serves correct). If you squandered it all on the boss rematches and the fortress bosses, then Crash Bomb is the most powerful option. However, consider...
Wily Ship Second Phase - Crash Bomb is the only true weakness at quadruple damage. All other effective weapons (Metal Blade and Quick Boomerang, to my recollection) only do the same damage as your Mega Buster, so you'd probably save it for this.
Alien - The only weapon that works is Bubble Lead.