Canon (not to be confused with can(n)on) is a term used to designate what is considered "official" in a certain fictional universe. Unless otherwise specified in the OP, only canon material can be used in a debate thread. Non-canon material doesn't count - it never happened.
Many debates have been held on exactly what is and is not canon.
The generally agreed-upon definition is that the work by the original author and creator of the fictional setting is canon, unless the author or the copyright holder declares otherwise.
So for most manga series, that means the original manga is canon, while the anime is not (since the anime is simply an adaptation of the manga made by others). Databooks are considered secondary canon since scans tend to contradict them or the writer is a moron (Haku is FTL...) It should be noted that this is often evaluated on a case - by - case basis. A notable exception is the Tenchi Muyo! series, where the OVAs are canon since they are the original work of the author, and the manga is written by a completely different person and is not canon.
Major canon debates that have occurred in the OBD:
1. Dragon Ball. Despite the overwhelming view that the manga (and now, Dragon Ball Super) is the only canon, many people (specifically the Phenom Brigade), have attempted in the past to claim that the anime, movies, and GT are canon. The movies are obviously non - canon, since they fit nowhere in the timeline (Movie 9 is debatable). GT was made after Toriyama quit, and was created by different people with no basis on the manga. Also, there were many irreconcilable contradictions with the manga, so it is generally considered to be non - canon.
The anime is sometimes argued to be canon, usually by using various Toriyama interviews. However, none of these interviews actually state the anime is canon, just that Toriyama contributed some character designs for GT and the filler anime. In fact, they often provide evidence that it is not canon, by explaining how Toriyama would just let the animation studio do whatever they wanted after giving them a rough outline, or pointing out that there were many differences between the anime and the manga.
The recent TV special "Yo! Son Goku and his Friends Return!" has been argued to be canon, because it was written directly by Toriyama. However, there is little in it that is usable for debate purposes, so there have not been many major arguments over it.
Recently, it has been argued that the backstory of the upcoming MMORPG Dragonball Online is canon, due to Toriyama's involvement. This is still disputed. A similar claim is being made about a recent spin off manga about Bardock. Adding to the confusion, it has been argued that due to the nature of time travel in Dragon Ball, most of the Bardock spinoff would technically take place in an alternate time line, much like Future Trunks.
2. Star Wars. The Star Wars Expanded Universe used to be canon, based on George Lucas' subordinate Leeland Chee, who developed the "canon holocron" which explained the different levels of canon. However, with the purchase of the franchise by Disney, the EU was purged and made into a separate, non-canon Legends timeline.
3. Neon Genesis Evangelion. A large number of secondary sources, some of which seem to be contradictory, say that the anime is canon since it is the original work of Anno. While the manga was done by a different author; Anno stated that the Manga is its own canonicity and the Manga can be used to support the anime when they are in explicit agreement; while the Rebuild of Evangelion series classifies as a wholly separate continuity with its own backstory, which may not be applicable to the TV series and shall be treated as such. Their also the deleted scenes that appear in the director's cut episodes but they are often not counted as canon since they where removed on final or official versions and take precedence over the TV version in the event of any theoretical contradiction.
4. Transformers. Everything is canon. In Transformers, "canon" is for all intents and purposes a synonym for "official". If it was released by a Transformers licenser with Hasbro approval, then it is canonical. However, simply being canonical doesn't say anything about what continuity or continuities it applies to.
3H and IDW have both released comics which take place "just offscreen" during the Beast Wars cartoon, but these comics contradict each other. Rather than the newer IDW comics invalidating the 3H story or retconning it out of existence, the two are simply relegated to separate but closely parallel universes. While this "multiverse" approach helps to ensure that essentially all Transformers fiction is given a certain amount of validity, there are occasions when individual texts within the same continuity contradict each other. (The history of the Constructicons in the original cartoon series is probably the most infamous example.) When this happens, there is no clear way to proceed. Fans may reach a consensus on how to best interpret the evidence, but this consensus is not official and therefore not canon. (Until/unless somebody writing official fiction drops it in, transforming it from fanon to canon.) Each fan's interpretation of such events constitutes a part of their personal canon, a subjective collection of ideas about the official fiction which is typically a work in progress that is constantly being reevaluated.
5. Doctor Who. Much like Star Wars, there is a canon hierarchy, although it is not as official. The primary canon consists of:
- The main Doctor Who television series (including the Christmas specials and other special episodes that tie into the main plot, including most of the recent Comic Relief specials) - The 1996 TV movie - The spinoff series Sarah Jane Adventures, Torchwood, and K9, as well as the K9 and Company pilot (ugh...)
The secondary canon consists of most other licensed media, including novels, comic books, video games, etc.