Hello, Lady Evenstar!
"Mary-Sue" is a name to describe a certain kind of original character that is found in fanfiction. She is the way the author sub-consciously writes herself into the story. (Self-inserts of the "author falls into Middle Earth" variety are related.)It's something that most fanfiction authors do, especially when they're starting out.
Mary-Sues share certain characteristics. Because authors are living out their own dreams via their original character, they tend to make the "Mary-Sue" character as fun and interesting for themselves as they can. This makes Mary-Sue characters rather easy to spot.
Mary-Sue tends to be:
Very beautiful (or ugly except to the hero). She often has odd coloring, for example, white, silver or red hair, green or mismatched eyes (one blue, one brown, etc). She is usually tall and striking.
Fantasy Mary-Sues often have nouns for names, for example: Cat, Raven, Rowan, Neko (Cat), Kitsune (Fox), and so on.
Mary-Sue often has hidden or arcane talents. She may have precognition, the ability to cast spells or communicate with people or animals telepathically. (Wolves, cats, dragons and horses are favorite Mary-Sue companions.)
Mary-Sue is either related to a famous canon character or is his love interest -- possibly both. For example, Mary-Sue may be Legolas' long lost twin sister who is so enchanting that Aragorn dumps Arwen to marry her. Twins and bastard half-sisters are common Mary-Sues.
Mary-Sue is usually liked, admired or adored by all canon characters within seconds of meeting them. They feel some "strange attraction" to her or intuit that she is very "important" or has "latent powers". Gandalf may realize that she holds they key to saving Middle Earth, the Ringbearer, etc. Mary-Sue is the most intelligent, most sensitive, most fun person they have ever met. She also knows more about archery than Legolas, can outride Eomer, and can teach Aragorn a thing or two about swordfighting.
Mary-Sue often has a tragic past or future. Her family was killed by Orcs, she was a slave of the Easterlings, and so on. At the end of the story she may die of some illness or in battle, so that the canon characters can weep over her and vow never to love another. Their lives will never be the same without her, etc.
Why do people revile Mary-sue? It's because she's not very interesting to anyone other than the author. No one else empathizes with her the way the author does, and so, instead of identifying with her, the readers find her boring, or annoying, or both. That is not to say there aren't good, well-written "Mary-Sue" stories out there. Some of them are quite good. It's just that the majority of them aren't, so readers learn to avoid Mary-Sue stories if they can.
If someone calls your story a Mary-Sue, it doesn't mean you're a bad writer. It just means that you've started out by writing a wish-fulfilment story. That's fine. But people are more likely to read and enjoy your stories that are not Mary-sues.
You can learn more about Mary-Sue at this site:
Mary Sue Society
They have litmus tests there so you can see for yourself if your story qualifies as a Mary-sue. Their essays are also quite amusing.
Cheers!
-Lily