If this thread's title doesn't specify it's spoilery, don't bring any up.
Discontinuities are narrative problems in a particular story, which we can solve only through speculation. They are to be contrasted with production errors, which have to do with technical mistakes made by the production crew in the assembly of the story in question.
The discontinuity index gives us the ability to have free-ranging discussions about these apparent errors, without fear of cluttering the otherwise factual story pages with our own pet theories. At the top right of this page, you will find a guide to help you locate the particular story you wish to discuss.
Discontinuity
"Discontinuity" may be thought of as elements of one story which seem to be in conflict with other stories. For instance, the explanation of the final days of planet Earth given in The End of the World seem to be opposed to those seen in The Ark, because the sun is described as having essentially exploded at two radically different points in history. We can make a simple note of that fact in the "Behind the scenes" section on the story page, but explanations of how the discontinuity is to be resolved belong in the pages that branch off from this one.
The reason this is a matter for discussion, rather than an absolute factual "error", is that the Doctor, as a time traveler, has the ability to change history. Likewise the Time Lords, as a consequence of their war with the Daleks could well have radically altered history upon their eradication. The Meddling Monk, Master, Drax, Rani, Romana or any other rogue Time Lord could have changed history as well. It is further possible, in this instance, that the people aboard the Ark may simply have been wrong about their assessment of the sun's condition. There is no one, absolute account which has ever been offered by Doctor Who fiction to explain the apparent contradiction.
There are probably two good guidelines for determining whether the point you wish to include belongs here. First, if your explanation goes well outside what is actually offered in a story, it almost certainly belongs here. The chances are that if you've had to go beyond the narrative to find an answer for the element of discontinuity, there are other points of view possible. Hence, the point belongs here, where a discussion can ensue, rather than on a story page, where discussion is not allowed. Second, if the issue that concerns you has not achieved wide, cite-able debate in multiple professional publications, it also belongs here. Note the importance of multiple, professional sources. An off-hand one-liner in an article in Doctor Who Magazine isn't enough to qualify an element of discontinuity for inclusion in the main article space. Likewise, the sources must be professional publications, not blogs or forum pages. In order to qualify for its own page within the main article space, it must have generated significant and wide-ranging output across several, highly-regarded sources. Preferably, these sources should themselves have their own articles on this wiki. In truth, the only element of discontinuity that currently rises to this standard is that surrounding the UNIT dating controversy, because it has been often noted by many professional writers, and has even been the focus of academic essays in analytical reference works.
Plot holes
A "plot hole" is a narrower kind of discontinuity. It's narrative problem that affects only the one story at issue. For instance, if the Doctor says in one part of the story that X, Y, and Z are required to resolve the plot, but in the end is seen only providing X and Y, then there is a plot hole because he didn't address Z. A recent example might be seen in Flesh and Stone, where Amy, the Eleventh Doctor and the clerics all understand that Amy can only open her eyes for a second, but yet she's clearly shown to have her eyes open for several seconds when looking at the Time Field. There's no clear reason for why she was able to violate what the story had set up, so the point belongs here, where it can be discussed.