Don Corneo is the epitome of scum. When you reach Wall Market early in the game, you'll find out that he is the owner of a not-so-elegant establishment called the Honey Bee Inn. He also kidnaps girls so he can determine which one will be his "lucky" companion for the night. It just so happens that he's kidnapped Tifa. After Tifa is rescued, you think you've seen the last of him, but not quite. He pops up again later in Wutai, THIS TIME, capturing Yuffie and Elena (the female member of The Turks).
Dyne used to be Barret's best friend, until Shinra burned down the original North Corel. Like Barret, he has a cybernetic arm where he can install guns. Unlike Barret, he has no reservations as to what and whom he shoots with it. Having gone completely crazy, he's taken over the Corel Prison and plans to kill as many people as he can because he just doesn't care anymore. So, Barret has to take out his old friend one-on-one. This is an easy fight. Equip Barret with your best stuff (Enemy Skills, summons, Time, Restore, etc.). Also use a Hyper on him to make his Limits go up quicker. Just keep blasting away at Dyne with Limit Breaks and summons.
Never in all the RPG's I've played have I encountered a boss as easy as this guy. Okay, a little background info, first. Long ago, Cosmo Canyon was invaded by a tribe of warriors called the Gi. Red XIII's father played a major role in fighting the Gi. When Bugenhagen opens the Gi Cave to show Red XIII what really happened to his father (Seto), the ghosts of the dead Gi attack. At the end of the cave, just before the final fate of Seto is revealed, Gi Nanataku attacks you. Now there are some things to remember about undead creatures in RPG's. They're usually not only weak against Fire, but also against HEALING SPELLS. You can fight him normally if you want, but this is a long game and you probably don't want to dawdle. Give Gi Nanataku an X-potion. Normally, it completely heals one of your own characters. But it'll instantly KILL Gi Nanataku.
There's not a lot of background behind this monster. He's in the caves in Mt. Nibel and he blocks your path to Rocket Town. He doesn't seem to have any connection to Shinra or Sephiroth, so I included him here. To beat him, use the usual boss strategies, EXCEPT don't use Fire, Ifrit, or Vincent's Level 1 Limit break. Fire-based attacks wil heal him. But feel free to use whatever else is at your disposal. Use Big Guard early during the battle. When he casts Trine, the person with the Enemy Skills will add it to the Enemy Skills collection.