Character error
Jasper acts as if blind in some episodes but can clearly see in others.
When Homer is Pie Man, Mr Burns unmasks him, recognizes him and remembers his name.
Continuity
Ralph Wiggum is in Bart's class in some episodes and Lisa's in others.
The interior of the family's home changes from time to time, e.g. the size of the archway and what room it leads to.
Springfield Elementary School is yellow in some episodes and
white in others, sometimes changing color multiple times in the same episode (later episodes corrected this continuity problem by making the school's official color yellow, and including a picture of the "old, white" school in Principal Skinner's office.)
In a few episodes, the twins, who are supposed to be sheet white, are yellow in brief scenes.
When the older, male judge character is introduced in The Simpsons: Bart Gets Hit by a Car, he is referred to as Judge Moulton. In all other appearances and source books he is called Judge Snyder.
When Milhouse Van Houten first appears in The Simpsons: Simpsons Roasting on an Open Fire, he has black hair. In all subsequent episodes, he has blue hair.
Chief Wiggum's proper hair color is blue. A common animation error is for it to be black in certain scenes (this generally happens in night scenes, or other scenes in dark places.)
Some characters are declared dead in one episode, then appear alive in a later episode without explanation or a humorous purpose.
Waylon Smithers was designed to be "Mister Burns' white sycophant," as it "would be a bad idea to have a black subservient character" putting up with the abuse that Burns doles out. (Quotes are from David Silverman, one of the show's main producers.) Ironically, Smithers is colored as an African-American in his first appearance The Simpsons: Homer's Odyssey, due to color stylist Gyorgyi Peluce's misinterpretation of the new character. The mistake was quickly caught and corrected starting with the immediate next episode The Simpsons: There's No Disgrace Like Home.
Although the giant stone head in the Simpson basement is gray, in some episodes it has appeared rainbow-colored.
Some characters have brown skin in some episodes and yellow skin in others. Two examples would be Hans Moleman and Judge Snyder.
When wearing her normal green dress, Marge wears a red necklace. However, often she is shown for a single frame or two wearing a white one (generally this happens in close-up shots of her head.) This happens in numerous episodes.
Many students at Springfield Elementary swap between Bart's and Lisa's class.
In the intro sequence, Lisa is playing a blue sax, but during the actual show she has a golden sax (this was corrected in the new HD intro).
When Moe appears in the first season he has black hair. In all subsequent seasons his hair is a grayish-blue.
In many episodes, the locations of buildings in Springfield move around.
Mr Burns shocks Krusty with a Cattle Prod but it doesn't affect Krusty's pacemaker.
Incorrectly regarded as goofs
"In one episode they say this, but then in another episode they say that, and in yet another episode they say the other." As this is an animated comedy series, the emphasis is clearly on laughs rather than complete verisimilitude. Efforts are certainly made to create a vaguely consistent setting in which mostly consistent characters live and work, and many episodes refer to each other, but rigid consistency of every single detail in all episodes is unnecessary. In many episodes, the fact that something is inconsistent is the express point of a gag. Our general rule is that each episode is expected to be consistent within itself, but intra-episode inconsistencies are not being listed. There can be exceptions for unusually noteworthy matters, e.g., inconsistencies repeated in multiple episodes (such as the hair and skin colors of secondary characters, and the layouts of the main landmarks), drastic changes to a character's nature (such as Ralphie's school status, Jasper's abilities, or Milhouse's hair color), or something with an interesting anecdote behind it (such as Smithers' skin color).
Fat Tony's real name changes from episode to episode, but this is probably intentional because it is common for Mafia hit men (in popular culture, and possibly in reality as well) to have many different identities.
Plot holes
The family portraits in the Hibbert home have yellow skin.