The most frequent animation error seen in the show is the appearance of the eyebrows (and some cases the eyes). The eyebrows are known to disappear and/or reappear at times, or in rare cases even "duplicate", creating an extra pair of eyebrows (as seen on Lois' father when Peter meets him in "Death Lives").
When shown a quick view of Cleveland's House it is one and a half story Cape Cod house, like Peter's House. But, whenever Peter does one of his "Shenanigans' near Cleveland's house, it is a full two story, non-Cape Cod home.
In "No Chris Left Behind", the Morningwood Academy sign has "NO IRISH" crossed out and "IRISH OK" written under it. The next time we see the sign, there is no trace whatsoever of either line of the Irish text.
Several times throughout the series, characters appear on game shows. But the versions of the game shows are outdated versions that no longer exist. For instance, the Griffins appeared on Family Feud, with its old set style and Richard Dawson hosting. Likewise, Peter appeared on Wheel of Fortune, which used its old "buy prizes after each round" format.
Peter lives in a Cape Cod house. They have one full floor and an upstairs which has slanted walls. When there are screen shots of Peter's room, Chris's room, Stewie or Meg's room, they have straight walls.
Peter's shirt buttons the wrong way. Men's shirts button with the left side over the right side, but Peter's shirt is backwards, in the same direction as Lois's shirt. Occasionally, but very rarely, this changes.