The giant eyeball is, as the name indicates, a giant eyeball encountered by an elderly couple around 30 years ago.
The eye somehow ended up in Lake Gravity Falls 30 years ago before the events of the series, wherein it surfaced shortly afterwards and was then spotted by an elderly couple. The couple panicked, ran away and had their memories erased by the Blind Eye Society; after the erase, they were no longer panicked. This all occured in a flashback within the episode "Society of the Blind Eye."
The giant eyeball is a seemingly disembodied eyeball of huge proportions.
- 207. "Society of the Blind Eye" (flashback)