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.
In R-Type 2, the Bydo Core is called Womb (but often mispelled as Woom in early official materials) and resembles a large crab hanging onto the wall with two Bydo embyos inside it. The R-Type Official Website states that this is a clone of the Bydo Core in R-Type. There are four other R-9s trapped by the Core. Though they can be freed by destroying the tethers and help provide distractions, they typically don't last long. In the SNES version, it's different, one of those R-9 fighters will free from destruction, allowing one to sacrifice itself. In the Dimensions Version it has the same as the SNES version compared to the arcade.
The Bydo Core is called the Amber Eye in this game. On the sides of the Amber Eye are plumes of miniature Ebon Eyes. Just like the previous Bydo Core, when the Amber Eye is defeated, it explodes and sucks the human fleet into it. The human fleet gets broken down by the Solar Envoy Guards.
The Bydo Core is sometimes called Bydo. In the Game Boy release of the original R-Type, it was called El Supremo (The Supreme) in the manual's story.
See video of R-Type's final boss fight on YouTube here).