The Bydo Core only refers to the unknown and mysterious secret bioweapon and evolving supercomputer leader that serves as the final boss of most of the R-Type games.
The Bydo Core was created by Humans in the 26th Century as the source of the Bydo creatures, who were also created by Humans in the same century. However, an error activated the Bydo Core too early in the Solar System and the creation attacked the humans. As a result, the humans banished the Bydo Core into another dimension where it began to evolve, grow, reproduce, and hold a grudge against its enemies. It then found a way to return to the Humans' dimension (in the 22nd Century), through a different solar system. The Bydo Core quickly assimilated the solar system it had exited the other dimension from, creating the Bydo Empire. The monstrous life-forms created by the Core then made a beeline to Earth, taking over everything in their way.
In the original R-Type, The Bydo Core looks like a green, wizened dwarf-like creature with an eye embedded in his forehead. A fleshy gateway separates the Core from the external world and it opens occasionally. Firing a Force into the Core while the portal is open is the best way to damage it. The trick is surviving long enough for the Force to deal lethal damage.
Bydo Dimension. It is called Mother Bydo in this game.
A Bydo created in the lab that repeatedly spews out all kinds of matter and energy. It can deliver any Bydo attack.
The Bydo Core is called the Ebon Eye in this mission. When defeated, the Ebon Eye's death throes pull in the Space Corps armada. Though a few squads of R-craft can be seen escaping, the majority of craft and all battleships are absorbed by the Bydo. The armada's commander survives as a residual intelligence, seemingly unaware that they've become Bydo. This unlocks the Bydo campaign.
Bydo? Ebon Eye -Eye of Darkness-
Source of the Bydo life forms. Surrounded by huge thrusters living (?) within its walls.
See video of R-Type's final boss fight on YouTube here).